data_source
stringclasses 6
values | prompt
listlengths 2
2
| ability
stringclasses 1
value | reward_model
dict | extra_info
dict | problem
stringlengths 15
4.65k
| prompt_messages
listlengths 2
2
| gt
stringlengths 1
386
|
|---|---|---|---|---|---|---|---|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Calculate: $-2^2 \\times (-1)^{2015}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "4",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Calculate: $-2^2 \times (-1)^{2015}$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Calculate: $-2^2 \\times (-1)^{2015}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
4
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "In the arithmetic sequence $\\{a_{n}\\}$, $a_{7}=4$, and $a_{19}=2a_{9}$.\n1. Find the general term formula for $\\{a_{n}\\}$.\n2. Let $b_{n}=\\frac{1}{2na_{n}}$. Find the sum of the first $n$ terms, $S_{n}$, of the sequence $\\{b_{n}\\}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac{n}{n+1}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
In the arithmetic sequence $\{a_{n}\}$, $a_{7}=4$, and $a_{19}=2a_{9}$.
1. Find the general term formula for $\{a_{n}\}$.
2. Let $b_{n}=\frac{1}{2na_{n}}$. Find the sum of the first $n$ terms, $S_{n}$, of the sequence $\{b_{n}\}$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "In the arithmetic sequence $\\{a_{n}\\}$, $a_{7}=4$, and $a_{19}=2a_{9}$.\n1. Find the general term formula for $\\{a_{n}\\}$.\n2. Let $b_{n}=\\frac{1}{2na_{n}}$. Find the sum of the first $n$ terms, $S_{n}$, of the sequence $\\{b_{n}\\}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{n}{n+1}
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Find all solutions to the inequality \\[\\frac{2x}{x-2} + \\frac{x-3}{3x} \\ge 2.\\] (Give your answer in interval notation.)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "(0, \\frac{5}{6}] \\cup (2, \\infty)",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Find all solutions to the inequality \[\frac{2x}{x-2} + \frac{x-3}{3x} \ge 2.\] (Give your answer in interval notation.)
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Find all solutions to the inequality \\[\\frac{2x}{x-2} + \\frac{x-3}{3x} \\ge 2.\\] (Give your answer in interval notation.) Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
(0, \frac{5}{6}] \cup (2, \infty)
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Find the inverse of the matrix\n\\[\n\\begin{pmatrix}\n5 & 15 \\\\\n2 & 6\n\\end{pmatrix}.\n\\]\nIf the inverse does not exist, then enter the zero matrix.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\begin{pmatrix} 0 & 0 \\\\ 0 & 0 \\end{pmatrix}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Find the inverse of the matrix
\[
\begin{pmatrix}
5 & 15 \\
2 & 6
\end{pmatrix}.
\]
If the inverse does not exist, then enter the zero matrix.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Find the inverse of the matrix\n\\[\n\\begin{pmatrix}\n5 & 15 \\\\\n2 & 6\n\\end{pmatrix}.\n\\]\nIf the inverse does not exist, then enter the zero matrix. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "What is the value of $d$ if $x\\cdot(4x-3)<d$ if and only when $x\\in \\left(-\\frac{5}{2},3\\right)$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "39",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
What is the value of $d$ if $x\cdot(4x-3)<d$ if and only when $x\in \left(-\frac{5}{2},3\right)$?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "What is the value of $d$ if $x\\cdot(4x-3)<d$ if and only when $x\\in \\left(-\\frac{5}{2},3\\right)$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
39
|
numina_olympiads
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given the area \\( t \\) of a triangle and one of its angles \\( C \\), what should be the lengths of the sides \\( a \\) and \\( b \\) passing through the vertex with angle \\( C \\) so that the side \\( c \\) opposite to angle \\( C \\) is minimized?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "c = 2 \\sqrt{t \\cdot \\tan \\frac{C}{2}}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given the area \( t \) of a triangle and one of its angles \( C \), what should be the lengths of the sides \( a \) and \( b \) passing through the vertex with angle \( C \) so that the side \( c \) opposite to angle \( C \) is minimized?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given the area \\( t \\) of a triangle and one of its angles \\( C \\), what should be the lengths of the sides \\( a \\) and \\( b \\) passing through the vertex with angle \\( C \\) so that the side \\( c \\) opposite to angle \\( C \\) is minimized? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
c = 2 \sqrt{t \cdot \tan \frac{C}{2}}
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "If the solution of the equation $\\frac{2x+m}{x-1}=1$ with respect to $x$ is negative, determine the range of values for $m$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "m > -1",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
If the solution of the equation $\frac{2x+m}{x-1}=1$ with respect to $x$ is negative, determine the range of values for $m$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "If the solution of the equation $\\frac{2x+m}{x-1}=1$ with respect to $x$ is negative, determine the range of values for $m$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
m > -1
|
numina_aops_forum
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "We define a beehive of order $n$ as follows:\n\na beehive of order 1 is one hexagon\n\nTo construct a beehive of order $n$ , take a beehive of order $n-1$ and draw a layer of hexagons in the exterior of these hexagons. See diagram for examples of $n=2,3$ Initially some hexagons are infected by a virus. If a hexagon has been infected, it will always be infected. Otherwise, it will be infected if at least 5 out of the 6 neighbours are infected.\n\nLet $f(n)$ be the minimum number of infected hexagons in the beginning so that after a finite time, all hexagons become infected. Find $f(n)$ . \n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "f(n) = 3n^2 - \\left\\lfloor \\frac{3}{2} n^2 + \\frac{3}{2} \\right\\rfloor",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
We define a beehive of order $n$ as follows:
a beehive of order 1 is one hexagon
To construct a beehive of order $n$ , take a beehive of order $n-1$ and draw a layer of hexagons in the exterior of these hexagons. See diagram for examples of $n=2,3$ Initially some hexagons are infected by a virus. If a hexagon has been infected, it will always be infected. Otherwise, it will be infected if at least 5 out of the 6 neighbours are infected.
Let $f(n)$ be the minimum number of infected hexagons in the beginning so that after a finite time, all hexagons become infected. Find $f(n)$ .
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "We define a beehive of order $n$ as follows:\n\na beehive of order 1 is one hexagon\n\nTo construct a beehive of order $n$ , take a beehive of order $n-1$ and draw a layer of hexagons in the exterior of these hexagons. See diagram for examples of $n=2,3$ Initially some hexagons are infected by a virus. If a hexagon has been infected, it will always be infected. Otherwise, it will be infected if at least 5 out of the 6 neighbours are infected.\n\nLet $f(n)$ be the minimum number of infected hexagons in the beginning so that after a finite time, all hexagons become infected. Find $f(n)$ . Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
f(n) = 3n^2 - \left\lfloor \frac{3}{2} n^2 + \frac{3}{2} \right\rfloor
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "The sequence $\\{a_n\\}$ is an arithmetic sequence with a non-zero common difference, and $S_n$ is the sum of its first $n$ terms. It is known that $a_2+a_3+a_5=20$, and $a_2$, $a_4$, $a_8$ form a geometric sequence. Let $M= \\frac{1}{S_1} + \\frac{1}{S_2} + \\ldots + \\frac{1}{S_n}$.\n\n(1) Find $M$; \n(2) The sum of the first $n$ terms of the sequence $\\{b_n\\}$ is $T_n$, and it is known that $T_n=2(b_n-1)$. Compare the size of $T_n$ and $M+1$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "T_n > M+1",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
The sequence $\{a_n\}$ is an arithmetic sequence with a non-zero common difference, and $S_n$ is the sum of its first $n$ terms. It is known that $a_2+a_3+a_5=20$, and $a_2$, $a_4$, $a_8$ form a geometric sequence. Let $M= \frac{1}{S_1} + \frac{1}{S_2} + \ldots + \frac{1}{S_n}$.
(1) Find $M$;
(2) The sum of the first $n$ terms of the sequence $\{b_n\}$ is $T_n$, and it is known that $T_n=2(b_n-1)$. Compare the size of $T_n$ and $M+1$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "The sequence $\\{a_n\\}$ is an arithmetic sequence with a non-zero common difference, and $S_n$ is the sum of its first $n$ terms. It is known that $a_2+a_3+a_5=20$, and $a_2$, $a_4$, $a_8$ form a geometric sequence. Let $M= \\frac{1}{S_1} + \\frac{1}{S_2} + \\ldots + \\frac{1}{S_n}$.\n\n(1) Find $M$; \n(2) The sum of the first $n$ terms of the sequence $\\{b_n\\}$ is $T_n$, and it is known that $T_n=2(b_n-1)$. Compare the size of $T_n$ and $M+1$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
T_n > M+1
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given a line $l: x - y + 4 = 0$ and a circle $C: \\begin{cases} x = 1 + 2\\cos\\theta \\\\ y = 1 + 2\\sin\\theta \\end{cases}$, find the minimum distance from any point on the circle $C$ to the line $l$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "2\\sqrt{2} - 2",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given a line $l: x - y + 4 = 0$ and a circle $C: \begin{cases} x = 1 + 2\cos\theta \\ y = 1 + 2\sin\theta \end{cases}$, find the minimum distance from any point on the circle $C$ to the line $l$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given a line $l: x - y + 4 = 0$ and a circle $C: \\begin{cases} x = 1 + 2\\cos\\theta \\\\ y = 1 + 2\\sin\\theta \\end{cases}$, find the minimum distance from any point on the circle $C$ to the line $l$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
2\sqrt{2} - 2
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given the function $f(x)=\\sin (2x+φ)$, where $φ$ is a real number. If $f(x)\\leqslant \\left| f\\left( \\frac{π}{6} \\right) \\right|$ holds for all $x\\in R$, and $f\\left( \\frac{π}{2} \\right) > f(π)$, determine the interval where $f(x)$ is monotonically increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\left[kπ+ \\frac{π}{6},kπ+ \\frac{2π}{3}\\right](k∈Z)",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given the function $f(x)=\sin (2x+φ)$, where $φ$ is a real number. If $f(x)\leqslant \left| f\left( \frac{π}{6} \right) \right|$ holds for all $x\in R$, and $f\left( \frac{π}{2} \right) > f(π)$, determine the interval where $f(x)$ is monotonically increasing.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given the function $f(x)=\\sin (2x+φ)$, where $φ$ is a real number. If $f(x)\\leqslant \\left| f\\left( \\frac{π}{6} \\right) \\right|$ holds for all $x\\in R$, and $f\\left( \\frac{π}{2} \\right) > f(π)$, determine the interval where $f(x)$ is monotonically increasing. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\left[kπ+ \frac{π}{6},kπ+ \frac{2π}{3}\right](k∈Z)
|
numina_olympiads
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Three clients are at the hairdresser, each paying their bill at the cash register.\n\n- The first client pays the same amount that is in the register and takes 10 reais as change.\n- The second client performs the same operation as the first.\n- The third client performs the same operation as the first two.\n\nFind the initial amount of money in the cash register, knowing that at the end of the three operations, the cash register is empty.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "8.75",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Three clients are at the hairdresser, each paying their bill at the cash register.
- The first client pays the same amount that is in the register and takes 10 reais as change.
- The second client performs the same operation as the first.
- The third client performs the same operation as the first two.
Find the initial amount of money in the cash register, knowing that at the end of the three operations, the cash register is empty.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Three clients are at the hairdresser, each paying their bill at the cash register.\n\n- The first client pays the same amount that is in the register and takes 10 reais as change.\n- The second client performs the same operation as the first.\n- The third client performs the same operation as the first two.\n\nFind the initial amount of money in the cash register, knowing that at the end of the three operations, the cash register is empty. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
8.75
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "In square $ABCD$, the coordinates of $A$ and $C$ are $\\left(1,-2\\right)$ and $\\left(4,1\\right)$, respectively. $AB$ is parallel to the $x$-axis. Find the coordinates of point $B$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "(4,-2)",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
In square $ABCD$, the coordinates of $A$ and $C$ are $\left(1,-2\right)$ and $\left(4,1\right)$, respectively. $AB$ is parallel to the $x$-axis. Find the coordinates of point $B$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "In square $ABCD$, the coordinates of $A$ and $C$ are $\\left(1,-2\\right)$ and $\\left(4,1\\right)$, respectively. $AB$ is parallel to the $x$-axis. Find the coordinates of point $B$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
(4,-2)
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given the function $f(x)=x^{3}+ax^{2}+bx+c$ which attains its minimum value of $-5$ at $x\\_0$, and its derivative $y=f′(x)$ passes through the points $(0,0)$ and $(2,0)$.\n(1) Find the values of $a$ and $b$;\n(2) Find $x\\_0$ and the expression of the function $f(x)$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "f(x)=x^{3}-3x^{2}-1",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given the function $f(x)=x^{3}+ax^{2}+bx+c$ which attains its minimum value of $-5$ at $x\_0$, and its derivative $y=f′(x)$ passes through the points $(0,0)$ and $(2,0)$.
(1) Find the values of $a$ and $b$;
(2) Find $x\_0$ and the expression of the function $f(x)$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given the function $f(x)=x^{3}+ax^{2}+bx+c$ which attains its minimum value of $-5$ at $x\\_0$, and its derivative $y=f′(x)$ passes through the points $(0,0)$ and $(2,0)$.\n(1) Find the values of $a$ and $b$;\n(2) Find $x\\_0$ and the expression of the function $f(x)$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
f(x)=x^{3}-3x^{2}-1
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Compute $\\gcd(97^{10} + 1, 97^{10} + 97^3 + 1)$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "1",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Compute $\gcd(97^{10} + 1, 97^{10} + 97^3 + 1)$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Compute $\\gcd(97^{10} + 1, 97^{10} + 97^3 + 1)$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
1
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "The coordinates of the point symmetric to point P(1, 1, -2) with respect to the xoy plane are __________.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "(1, 1, 2)",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
The coordinates of the point symmetric to point P(1, 1, -2) with respect to the xoy plane are __________.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "The coordinates of the point symmetric to point P(1, 1, -2) with respect to the xoy plane are __________. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
(1, 1, 2)
|
numina_synthetic_amc
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given an inverted cone with a base radius of $15 \\mathrm{cm}$ and height of $25 \\mathrm{cm}$ is filled with water, which is then poured into a cylinder with a horizontal base radius of $18 \\mathrm{cm}$, calculate the height in centimeters of the water in the cylinder.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "5.8",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given an inverted cone with a base radius of $15 \mathrm{cm}$ and height of $25 \mathrm{cm}$ is filled with water, which is then poured into a cylinder with a horizontal base radius of $18 \mathrm{cm}$, calculate the height in centimeters of the water in the cylinder.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given an inverted cone with a base radius of $15 \\mathrm{cm}$ and height of $25 \\mathrm{cm}$ is filled with water, which is then poured into a cylinder with a horizontal base radius of $18 \\mathrm{cm}$, calculate the height in centimeters of the water in the cylinder. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
5.8
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Find all values of \\(x\\) that satisfy \\[\\frac{x^2}{x+2} \\ge \\frac{3}{x-2} + \\frac{7}{4}.\\]\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "(-2, 2) \\cup [3, \\infty)",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Find all values of \(x\) that satisfy \[\frac{x^2}{x+2} \ge \frac{3}{x-2} + \frac{7}{4}.\]
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Find all values of \\(x\\) that satisfy \\[\\frac{x^2}{x+2} \\ge \\frac{3}{x-2} + \\frac{7}{4}.\\] Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
(-2, 2) \cup [3, \infty)
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given a set $M$ that satisfies $\\{1, 2\\} \\subseteq M \\subset \\{1, 2, 3, 4\\}$, find the number of possible sets for $M$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "3",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given a set $M$ that satisfies $\{1, 2\} \subseteq M \subset \{1, 2, 3, 4\}$, find the number of possible sets for $M$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given a set $M$ that satisfies $\\{1, 2\\} \\subseteq M \\subset \\{1, 2, 3, 4\\}$, find the number of possible sets for $M$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
3
|
numina_olympiads
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "The edges of a rectangular prism are 26, 20, and 8 units long. Build straight gable roofs above the $26 \\times 8$-us faces, where the \"slanted\" edges are 13 units long, and the \"horizontal\" edge is 20 units long. What is the volume of the part of the rectangular prism obtained by removing the mirror images, with respect to the faces to which the roofs are attached, of the roofs from the prism?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "1911.76 \\text{ cubic units}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
The edges of a rectangular prism are 26, 20, and 8 units long. Build straight gable roofs above the $26 \times 8$-us faces, where the "slanted" edges are 13 units long, and the "horizontal" edge is 20 units long. What is the volume of the part of the rectangular prism obtained by removing the mirror images, with respect to the faces to which the roofs are attached, of the roofs from the prism?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "The edges of a rectangular prism are 26, 20, and 8 units long. Build straight gable roofs above the $26 \\times 8$-us faces, where the \"slanted\" edges are 13 units long, and the \"horizontal\" edge is 20 units long. What is the volume of the part of the rectangular prism obtained by removing the mirror images, with respect to the faces to which the roofs are attached, of the roofs from the prism? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
1911.76 \text{ cubic units}
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Acute-angled $\\triangle ABC$ is inscribed in a circle with center at $O$; $\\stackrel \\frown {AB} = 100^\\circ$ and $\\stackrel \\frown {BC} = 80^\\circ$.\nA point $E$ is taken in minor arc $AC$ such that $OE$ is perpendicular to $AC$. The task is to determine the ratio of the magnitudes of $\\angle OBE$ and $\\angle BAC$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac{1}{2}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Acute-angled $\triangle ABC$ is inscribed in a circle with center at $O$; $\stackrel \frown {AB} = 100^\circ$ and $\stackrel \frown {BC} = 80^\circ$.
A point $E$ is taken in minor arc $AC$ such that $OE$ is perpendicular to $AC$. The task is to determine the ratio of the magnitudes of $\angle OBE$ and $\angle BAC$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Acute-angled $\\triangle ABC$ is inscribed in a circle with center at $O$; $\\stackrel \\frown {AB} = 100^\\circ$ and $\\stackrel \\frown {BC} = 80^\\circ$.\nA point $E$ is taken in minor arc $AC$ such that $OE$ is perpendicular to $AC$. The task is to determine the ratio of the magnitudes of $\\angle OBE$ and $\\angle BAC$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{1}{2}
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "If the curve $y= \\frac {1}{2e}x^2$ and the curve $y=alnx$ have a common tangent at their common point $P(s, t)$, then the real number $a=$ ______.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "1",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
If the curve $y= \frac {1}{2e}x^2$ and the curve $y=alnx$ have a common tangent at their common point $P(s, t)$, then the real number $a=$ ______.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "If the curve $y= \\frac {1}{2e}x^2$ and the curve $y=alnx$ have a common tangent at their common point $P(s, t)$, then the real number $a=$ ______. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
1
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given triangle $ABC$, with $a=3 \\sqrt {3}$, $c=2$, and $B=150^{\\circ}$, find $b$ and the area of the triangle $S_{\\triangle ABC}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac{3}{2} \\sqrt {3}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given triangle $ABC$, with $a=3 \sqrt {3}$, $c=2$, and $B=150^{\circ}$, find $b$ and the area of the triangle $S_{\triangle ABC}$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given triangle $ABC$, with $a=3 \\sqrt {3}$, $c=2$, and $B=150^{\\circ}$, find $b$ and the area of the triangle $S_{\\triangle ABC}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{3}{2} \sqrt {3}
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "If all the sample points in a scatter plot are on the same line, then what are the respective values of the sum of squared residuals and the correlation coefficient?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "0,1",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
If all the sample points in a scatter plot are on the same line, then what are the respective values of the sum of squared residuals and the correlation coefficient?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "If all the sample points in a scatter plot are on the same line, then what are the respective values of the sum of squared residuals and the correlation coefficient? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
0,1
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Compute $\\sin 135^\\circ$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac{\\sqrt{2}}{2}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Compute $\sin 135^\circ$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Compute $\\sin 135^\\circ$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{\sqrt{2}}{2}
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given that $\\{a_n\\}$ is a monotonically increasing arithmetic sequence with the first term $a_1=3$ and the sum of the first $n$ terms denoted as $S_n$, and $\\{b_n\\}$ is a geometric sequence with the first term $b_1=1$ and such that $a_2b_2=12$ and $S_3+b_2=20$.\n\n(Ⅰ) Find the general formulas for $\\{a_n\\}$ and $\\{b_n\\}$.\n(Ⅱ) Let $C_n=n\\cdot b_n$ for $n\\in\\mathbb{N}^+$, find the sum of the first $n$ terms of $\\{c_n\\}$, denoted as $T_n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "(n-1)\\cdot 2^n+1",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given that $\{a_n\}$ is a monotonically increasing arithmetic sequence with the first term $a_1=3$ and the sum of the first $n$ terms denoted as $S_n$, and $\{b_n\}$ is a geometric sequence with the first term $b_1=1$ and such that $a_2b_2=12$ and $S_3+b_2=20$.
(Ⅰ) Find the general formulas for $\{a_n\}$ and $\{b_n\}$.
(Ⅱ) Let $C_n=n\cdot b_n$ for $n\in\mathbb{N}^+$, find the sum of the first $n$ terms of $\{c_n\}$, denoted as $T_n$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given that $\\{a_n\\}$ is a monotonically increasing arithmetic sequence with the first term $a_1=3$ and the sum of the first $n$ terms denoted as $S_n$, and $\\{b_n\\}$ is a geometric sequence with the first term $b_1=1$ and such that $a_2b_2=12$ and $S_3+b_2=20$.\n\n(Ⅰ) Find the general formulas for $\\{a_n\\}$ and $\\{b_n\\}$.\n(Ⅱ) Let $C_n=n\\cdot b_n$ for $n\\in\\mathbb{N}^+$, find the sum of the first $n$ terms of $\\{c_n\\}$, denoted as $T_n$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
(n-1)\cdot 2^n+1
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Calculate the value of $\\frac{-2 \\sqrt{3}+i}{1+2 \\sqrt{3}i}+\\left( \\frac{ \\sqrt{2}}{1-i} \\right)^{2017}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac{ \\sqrt{2}}{2}+\\left( \\frac{ \\sqrt{2}}{2}+1\\right)i",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Calculate the value of $\frac{-2 \sqrt{3}+i}{1+2 \sqrt{3}i}+\left( \frac{ \sqrt{2}}{1-i} \right)^{2017}$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Calculate the value of $\\frac{-2 \\sqrt{3}+i}{1+2 \\sqrt{3}i}+\\left( \\frac{ \\sqrt{2}}{1-i} \\right)^{2017}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{ \sqrt{2}}{2}+\left( \frac{ \sqrt{2}}{2}+1\right)i
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Let $z$ be a complex number, and let $a(z)$ denote the smallest positive integer $n$ such that $z^n = 1$. Then, for the imaginary unit $i$, find $a(i)$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "4",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Let $z$ be a complex number, and let $a(z)$ denote the smallest positive integer $n$ such that $z^n = 1$. Then, for the imaginary unit $i$, find $a(i)$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Let $z$ be a complex number, and let $a(z)$ denote the smallest positive integer $n$ such that $z^n = 1$. Then, for the imaginary unit $i$, find $a(i)$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
4
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given two real numbers $r > 1$ and $s > 1$ such that $\\frac{1}{r} + \\frac{1}{s} = 1$ and $rs = 9$, find the value of $s$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac{9 + 3\\sqrt{5}}{2}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given two real numbers $r > 1$ and $s > 1$ such that $\frac{1}{r} + \frac{1}{s} = 1$ and $rs = 9$, find the value of $s$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given two real numbers $r > 1$ and $s > 1$ such that $\\frac{1}{r} + \\frac{1}{s} = 1$ and $rs = 9$, find the value of $s$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{9 + 3\sqrt{5}}{2}
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "A point is selected at random from the segment of the number line from 0 to 6. What is the probability that the point is closer to 5 than to 1?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "0.5",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
A point is selected at random from the segment of the number line from 0 to 6. What is the probability that the point is closer to 5 than to 1?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "A point is selected at random from the segment of the number line from 0 to 6. What is the probability that the point is closer to 5 than to 1? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
0.5
|
numina_olympiads
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Another trapezoid \\(ABCD\\) has \\(AD\\) parallel to \\(BC\\). \\(AC\\) and \\(BD\\) intersect at \\(P\\). If \\(\\frac{[ADP]}{[BCP]} = \\frac{1}{2}\\), find \\(\\frac{[ADP]}{[ABCD]}\\). (Here, the notation \\(\\left[P_1 \\cdots P_n\\right]\\) denotes the area of the polygon \\(P_1 \\cdots P_n\\)).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "3 - 2\\sqrt{2}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Another trapezoid \(ABCD\) has \(AD\) parallel to \(BC\). \(AC\) and \(BD\) intersect at \(P\). If \(\frac{[ADP]}{[BCP]} = \frac{1}{2}\), find \(\frac{[ADP]}{[ABCD]}\). (Here, the notation \(\left[P_1 \cdots P_n\right]\) denotes the area of the polygon \(P_1 \cdots P_n\)).
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Another trapezoid \\(ABCD\\) has \\(AD\\) parallel to \\(BC\\). \\(AC\\) and \\(BD\\) intersect at \\(P\\). If \\(\\frac{[ADP]}{[BCP]} = \\frac{1}{2}\\), find \\(\\frac{[ADP]}{[ABCD]}\\). (Here, the notation \\(\\left[P_1 \\cdots P_n\\right]\\) denotes the area of the polygon \\(P_1 \\cdots P_n\\)). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
3 - 2\sqrt{2}
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given a complex number $z$ that satisfies $\\dfrac{z}{z-i}=i$, find $\\overline{z}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac{1-i}{2}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given a complex number $z$ that satisfies $\dfrac{z}{z-i}=i$, find $\overline{z}$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given a complex number $z$ that satisfies $\\dfrac{z}{z-i}=i$, find $\\overline{z}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{1-i}{2}
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Find the domain of the rational function \\( f(x) = \\frac{x^3 - x^2 - 4x + 4}{x^2 - 5x + 6} \\). Express your answer as a union of intervals.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "(-\\infty, 2) \\cup (2, 3) \\cup (3, \\infty)",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Find the domain of the rational function \( f(x) = \frac{x^3 - x^2 - 4x + 4}{x^2 - 5x + 6} \). Express your answer as a union of intervals.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Find the domain of the rational function \\( f(x) = \\frac{x^3 - x^2 - 4x + 4}{x^2 - 5x + 6} \\). Express your answer as a union of intervals. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
(-\infty, 2) \cup (2, 3) \cup (3, \infty)
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "The equation $x^{2}-6x+8=0$ has two roots, which are the base and one of the legs of an isosceles triangle. The perimeter of this isosceles triangle is ______.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "10",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
The equation $x^{2}-6x+8=0$ has two roots, which are the base and one of the legs of an isosceles triangle. The perimeter of this isosceles triangle is ______.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "The equation $x^{2}-6x+8=0$ has two roots, which are the base and one of the legs of an isosceles triangle. The perimeter of this isosceles triangle is ______. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
10
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "In the expansion of $({3{x^3}-\\frac{1}{x}})^6$, the coefficient of $x^{2}$ is ____.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "135",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
In the expansion of $({3{x^3}-\frac{1}{x}})^6$, the coefficient of $x^{2}$ is ____.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "In the expansion of $({3{x^3}-\\frac{1}{x}})^6$, the coefficient of $x^{2}$ is ____. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
135
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Mark borrows $2000$ dollars from Jane, who charges an interest rate of $4\\%$ per month (compounds monthly). Determine the least integer number of months after which Mark will owe more than three times as much as he borrowed.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "30",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Mark borrows $2000$ dollars from Jane, who charges an interest rate of $4\%$ per month (compounds monthly). Determine the least integer number of months after which Mark will owe more than three times as much as he borrowed.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Mark borrows $2000$ dollars from Jane, who charges an interest rate of $4\\%$ per month (compounds monthly). Determine the least integer number of months after which Mark will owe more than three times as much as he borrowed. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
30
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Compute the sum of the series:\n\n\\[ 5(1+5(1+5(1+5(1+5(1+5(1+5(1+5(1+5(1+5(1+5(1+5)))))))))) \\]\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "305175780",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Compute the sum of the series:
\[ 5(1+5(1+5(1+5(1+5(1+5(1+5(1+5(1+5(1+5(1+5(1+5)))))))))) \]
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Compute the sum of the series:\n\n\\[ 5(1+5(1+5(1+5(1+5(1+5(1+5(1+5(1+5(1+5(1+5(1+5)))))))))) \\] Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
305175780
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given $\\cos \\alpha= \\frac {1}{7}$, $\\cos (\\alpha-\\beta)= \\frac {13}{14}$, and $0 < \\beta < \\alpha < \\frac {\\pi}{2}$. \nFind: \n(1) The value of $\\tan 2\\alpha$; \n(2) The magnitude of $\\beta$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac {\\pi}{3}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given $\cos \alpha= \frac {1}{7}$, $\cos (\alpha-\beta)= \frac {13}{14}$, and $0 < \beta < \alpha < \frac {\pi}{2}$.
Find:
(1) The value of $\tan 2\alpha$;
(2) The magnitude of $\beta$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given $\\cos \\alpha= \\frac {1}{7}$, $\\cos (\\alpha-\\beta)= \\frac {13}{14}$, and $0 < \\beta < \\alpha < \\frac {\\pi}{2}$. \nFind: \n(1) The value of $\\tan 2\\alpha$; \n(2) The magnitude of $\\beta$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac {\pi}{3}
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Compute the product of matrices $\\begin{pmatrix} 3 & 0 & 4\\\\ 2 & -1 & 0\\\\ 0 & 1 & -2 \\end{pmatrix}$ and $\\begin{pmatrix} 5 & -3 & 0 \\\\ 0 & 2 & 1 \\\\ 4 & 0 & -2 \\end{pmatrix}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\begin{pmatrix} 31 & -9 & -8 \\\\ 10 & -8 & -1 \\\\ -8 & 2 & 5 \\end{pmatrix}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Compute the product of matrices $\begin{pmatrix} 3 & 0 & 4\\ 2 & -1 & 0\\ 0 & 1 & -2 \end{pmatrix}$ and $\begin{pmatrix} 5 & -3 & 0 \\ 0 & 2 & 1 \\ 4 & 0 & -2 \end{pmatrix}$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Compute the product of matrices $\\begin{pmatrix} 3 & 0 & 4\\\\ 2 & -1 & 0\\\\ 0 & 1 & -2 \\end{pmatrix}$ and $\\begin{pmatrix} 5 & -3 & 0 \\\\ 0 & 2 & 1 \\\\ 4 & 0 & -2 \\end{pmatrix}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\begin{pmatrix} 31 & -9 & -8 \\ 10 & -8 & -1 \\ -8 & 2 & 5 \end{pmatrix}
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Calculate:<br/>$(1)\\frac{2}{3}+2023^{0}÷({\\frac{1}{3}})^{-1}$;<br/>$(2)x^{5}\\cdot x^{7}+x^{6}\\cdot (-x^{3})^{2}+2(x^{3})^{4}$;<br/>$(3)\\left(a+3b\\right)^{2}(a-3b)^{2}$;<br/>$(4)2023^{2}-2022\\times 2024$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "1",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Calculate:<br/>$(1)\frac{2}{3}+2023^{0}÷({\frac{1}{3}})^{-1}$;<br/>$(2)x^{5}\cdot x^{7}+x^{6}\cdot (-x^{3})^{2}+2(x^{3})^{4}$;<br/>$(3)\left(a+3b\right)^{2}(a-3b)^{2}$;<br/>$(4)2023^{2}-2022\times 2024$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Calculate:<br/>$(1)\\frac{2}{3}+2023^{0}÷({\\frac{1}{3}})^{-1}$;<br/>$(2)x^{5}\\cdot x^{7}+x^{6}\\cdot (-x^{3})^{2}+2(x^{3})^{4}$;<br/>$(3)\\left(a+3b\\right)^{2}(a-3b)^{2}$;<br/>$(4)2023^{2}-2022\\times 2024$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
1
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "It takes 20 minutes for Mark to walk one mile. At that rate, how far in miles will he walk in 45 minutes? Express your answer as a decimal to the nearest tenth.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "2.3 \\text{ miles}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
It takes 20 minutes for Mark to walk one mile. At that rate, how far in miles will he walk in 45 minutes? Express your answer as a decimal to the nearest tenth.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "It takes 20 minutes for Mark to walk one mile. At that rate, how far in miles will he walk in 45 minutes? Express your answer as a decimal to the nearest tenth. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
2.3 \text{ miles}
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "In $\\triangle ABC$, $BC=2 \\sqrt{5}$, $AC=2$, the area of $\\triangle ABC$ is $4$, then the length of $AB$ is _______.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "4\\sqrt{2}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
In $\triangle ABC$, $BC=2 \sqrt{5}$, $AC=2$, the area of $\triangle ABC$ is $4$, then the length of $AB$ is _______.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "In $\\triangle ABC$, $BC=2 \\sqrt{5}$, $AC=2$, the area of $\\triangle ABC$ is $4$, then the length of $AB$ is _______. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
4\sqrt{2}
|
numina_olympiads
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "The probability that all the digits in the four-digit number are different and the units digit is an odd number.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac{56}{225}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
The probability that all the digits in the four-digit number are different and the units digit is an odd number.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "The probability that all the digits in the four-digit number are different and the units digit is an odd number. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{56}{225}
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "What is the probability that Julia flips more heads than tails when she flips 12 coins?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac{3172}{8192}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
What is the probability that Julia flips more heads than tails when she flips 12 coins?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "What is the probability that Julia flips more heads than tails when she flips 12 coins? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{3172}{8192}
|
numina_synthetic_amc
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "What is the value of $\\frac{((2304 + 88) - 2400)^2}{121}$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac{64}{121}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
What is the value of $\frac{((2304 + 88) - 2400)^2}{121}$?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "What is the value of $\\frac{((2304 + 88) - 2400)^2}{121}$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{64}{121}
|
numina_synthetic_amc
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "A shuttle bus departs to a mountain resort every two hours starting at 8 AM and finishing at 6 PM. The bus carries 80 tourists on the 8 AM trip, and on each successive trip, it carries 3 tourists fewer than on the previous trip. How many tourists did the shuttle bus transport to the resort that day?\nA) 415\nB) 435\nC) 455\nD) 475\nE) 495\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "B) 435",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
A shuttle bus departs to a mountain resort every two hours starting at 8 AM and finishing at 6 PM. The bus carries 80 tourists on the 8 AM trip, and on each successive trip, it carries 3 tourists fewer than on the previous trip. How many tourists did the shuttle bus transport to the resort that day?
A) 415
B) 435
C) 455
D) 475
E) 495
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "A shuttle bus departs to a mountain resort every two hours starting at 8 AM and finishing at 6 PM. The bus carries 80 tourists on the 8 AM trip, and on each successive trip, it carries 3 tourists fewer than on the previous trip. How many tourists did the shuttle bus transport to the resort that day?\nA) 415\nB) 435\nC) 455\nD) 475\nE) 495 Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
B) 435
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given the set {x|x^{2}-3x-4 < 0} and N = {-2,-1,0,1,2}, determine the intersection of the set \\\\(M\\\\) and N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\{0,1,2\\}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given the set {x|x^{2}-3x-4 < 0} and N = {-2,-1,0,1,2}, determine the intersection of the set \\(M\\) and N.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given the set {x|x^{2}-3x-4 < 0} and N = {-2,-1,0,1,2}, determine the intersection of the set \\\\(M\\\\) and N. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\{0,1,2\}
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "How many of the divisors of $10!$ are larger than $9!$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "9",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
How many of the divisors of $10!$ are larger than $9!$?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "How many of the divisors of $10!$ are larger than $9!$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
9
|
numina_synthetic_amc
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "A Ferris wheel has a radius of 30 feet and revolves at a constant rate of one revolution per minute. Calculate how many seconds it takes a rider to travel from the bottom of the wheel to a point 15 vertical feet above the bottom.\nA) 10 seconds\nB) 15 seconds\nC) 20 seconds\nD) 25 seconds\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "C) 20",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
A Ferris wheel has a radius of 30 feet and revolves at a constant rate of one revolution per minute. Calculate how many seconds it takes a rider to travel from the bottom of the wheel to a point 15 vertical feet above the bottom.
A) 10 seconds
B) 15 seconds
C) 20 seconds
D) 25 seconds
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "A Ferris wheel has a radius of 30 feet and revolves at a constant rate of one revolution per minute. Calculate how many seconds it takes a rider to travel from the bottom of the wheel to a point 15 vertical feet above the bottom.\nA) 10 seconds\nB) 15 seconds\nC) 20 seconds\nD) 25 seconds Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
C) 20
|
numina_olympiads
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "\\[\n\\frac{x^{3}}{x^{3}+2 y^{2} z}+\\frac{y^{3}}{y^{3}+2 z^{2} x}+\\frac{z^{3}}{z^{3}+2 x^{2} y} \\geqslant 1\n\\]\n\nOn the table, there is a pile of \\( n \\) stones. In one move, any existing pile can be split into two smaller piles. At the same time, the sizes of any two piles on the table must be \"similar.\"\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "1",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
\[
\frac{x^{3}}{x^{3}+2 y^{2} z}+\frac{y^{3}}{y^{3}+2 z^{2} x}+\frac{z^{3}}{z^{3}+2 x^{2} y} \geqslant 1
\]
On the table, there is a pile of \( n \) stones. In one move, any existing pile can be split into two smaller piles. At the same time, the sizes of any two piles on the table must be "similar."
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "\\[\n\\frac{x^{3}}{x^{3}+2 y^{2} z}+\\frac{y^{3}}{y^{3}+2 z^{2} x}+\\frac{z^{3}}{z^{3}+2 x^{2} y} \\geqslant 1\n\\]\n\nOn the table, there is a pile of \\( n \\) stones. In one move, any existing pile can be split into two smaller piles. At the same time, the sizes of any two piles on the table must be \"similar.\" Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
1
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "A rectangle has vertices at \\((2, z), (10, z), (2, 5)\\), and \\((10, 5)\\). The area of the rectangle is 64 square units, and \\( z \\) is positive. Determine the value of \\( z \\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "13",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
A rectangle has vertices at \((2, z), (10, z), (2, 5)\), and \((10, 5)\). The area of the rectangle is 64 square units, and \( z \) is positive. Determine the value of \( z \).
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "A rectangle has vertices at \\((2, z), (10, z), (2, 5)\\), and \\((10, 5)\\). The area of the rectangle is 64 square units, and \\( z \\) is positive. Determine the value of \\( z \\). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
13
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "A and B play a guessing game where A thinks of a number denoted as $a$, and then B guesses the number A thought of, denoting B's guess as $b$. Both $a$ and $b$ belong to the set $\\{0,1,2,3,\\cdots,9\\}$. If $|a-b|\\leqslant 1$, then A and B are considered to have a \"telepathic connection\". Now, if two people play this game randomly, the probability that they have a \"telepathic connection\" is ______.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac{7}{25}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
A and B play a guessing game where A thinks of a number denoted as $a$, and then B guesses the number A thought of, denoting B's guess as $b$. Both $a$ and $b$ belong to the set $\{0,1,2,3,\cdots,9\}$. If $|a-b|\leqslant 1$, then A and B are considered to have a "telepathic connection". Now, if two people play this game randomly, the probability that they have a "telepathic connection" is ______.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "A and B play a guessing game where A thinks of a number denoted as $a$, and then B guesses the number A thought of, denoting B's guess as $b$. Both $a$ and $b$ belong to the set $\\{0,1,2,3,\\cdots,9\\}$. If $|a-b|\\leqslant 1$, then A and B are considered to have a \"telepathic connection\". Now, if two people play this game randomly, the probability that they have a \"telepathic connection\" is ______. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{7}{25}
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "In the xy-plane with a rectangular coordinate system, let vector $\\overrightarrow {a}$ = (cosα, sinα) and vector $\\overrightarrow {b}$ = (sin(α + π/6), cos(α + π/6)), where 0 < α < π/2.\n(1) If $\\overrightarrow {a}$ is parallel to $\\overrightarrow {b}$, find the value of α.\n(2) If tan2α = -1/7, find the value of the dot product $\\overrightarrow {a}$ • $\\overrightarrow {b}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac{\\sqrt{6} - 7\\sqrt{2}}{20}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
In the xy-plane with a rectangular coordinate system, let vector $\overrightarrow {a}$ = (cosα, sinα) and vector $\overrightarrow {b}$ = (sin(α + π/6), cos(α + π/6)), where 0 < α < π/2.
(1) If $\overrightarrow {a}$ is parallel to $\overrightarrow {b}$, find the value of α.
(2) If tan2α = -1/7, find the value of the dot product $\overrightarrow {a}$ • $\overrightarrow {b}$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "In the xy-plane with a rectangular coordinate system, let vector $\\overrightarrow {a}$ = (cosα, sinα) and vector $\\overrightarrow {b}$ = (sin(α + π/6), cos(α + π/6)), where 0 < α < π/2.\n(1) If $\\overrightarrow {a}$ is parallel to $\\overrightarrow {b}$, find the value of α.\n(2) If tan2α = -1/7, find the value of the dot product $\\overrightarrow {a}$ • $\\overrightarrow {b}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{\sqrt{6} - 7\sqrt{2}}{20}
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "A function $g$ has the properties that $g(4x) = 4g(x)$ for all positive real values of $x$, and that $g(x) = 2 - |x - 3|$ for $2 \\leq x \\leq 4$. Find the smallest $x$ for which $g(x) = g(8000)$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "3016",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
A function $g$ has the properties that $g(4x) = 4g(x)$ for all positive real values of $x$, and that $g(x) = 2 - |x - 3|$ for $2 \leq x \leq 4$. Find the smallest $x$ for which $g(x) = g(8000)$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "A function $g$ has the properties that $g(4x) = 4g(x)$ for all positive real values of $x$, and that $g(x) = 2 - |x - 3|$ for $2 \\leq x \\leq 4$. Find the smallest $x$ for which $g(x) = g(8000)$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
3016
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given sets $A \\cap B = \\{a, b\\}$ and $A \\cup B = \\{a, b, c, d\\}$, calculate the number of pairs of sets $A$ and $B$ that satisfy the above conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "4",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given sets $A \cap B = \{a, b\}$ and $A \cup B = \{a, b, c, d\}$, calculate the number of pairs of sets $A$ and $B$ that satisfy the above conditions.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given sets $A \\cap B = \\{a, b\\}$ and $A \\cup B = \\{a, b, c, d\\}$, calculate the number of pairs of sets $A$ and $B$ that satisfy the above conditions. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
4
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Let \\( g(x) = 3\\sin{x} + 4\\cos{x} + 2\\tan{x} + 5\\cot{x} \\). Considering \\( x \\) in radian measure, find the smallest positive value \\( s \\) for which \\( g(x) = 0 \\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "3",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Let \( g(x) = 3\sin{x} + 4\cos{x} + 2\tan{x} + 5\cot{x} \). Considering \( x \) in radian measure, find the smallest positive value \( s \) for which \( g(x) = 0 \).
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Let \\( g(x) = 3\\sin{x} + 4\\cos{x} + 2\\tan{x} + 5\\cot{x} \\). Considering \\( x \\) in radian measure, find the smallest positive value \\( s \\) for which \\( g(x) = 0 \\). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
3
|
numina_olympiads
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "The 50th term in the sequence \\( 5, 6x, 7x^{2}, 8x^{3}, 9x^{4}, \\ldots \\) can be found using the general term formula \\( a_n = (n+4)x^{n-1} \\). Find the 50th term.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "54x^{49}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
The 50th term in the sequence \( 5, 6x, 7x^{2}, 8x^{3}, 9x^{4}, \ldots \) can be found using the general term formula \( a_n = (n+4)x^{n-1} \). Find the 50th term.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "The 50th term in the sequence \\( 5, 6x, 7x^{2}, 8x^{3}, 9x^{4}, \\ldots \\) can be found using the general term formula \\( a_n = (n+4)x^{n-1} \\). Find the 50th term. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
54x^{49}
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "How many complex numbers $z$ such that $\\left| z \\right| < 20$ satisfy the equation\n\\[\n e^z = 1 - \\frac{z}{2} \\, ?\n\\]\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "1",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
How many complex numbers $z$ such that $\left| z \right| < 20$ satisfy the equation
\[
e^z = 1 - \frac{z}{2} \, ?
\]
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "How many complex numbers $z$ such that $\\left| z \\right| < 20$ satisfy the equation\n\\[\n e^z = 1 - \\frac{z}{2} \\, ?\n\\] Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
1
|
numina_olympiads
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given five points \\(A, B, C, D,\\) and \\(E\\) in the plane in general position (meaning no three of them are ever collinear, and no four of them are ever cocyclic), a separator is defined as a circle passing through three of these points such that one of the remaining two points lies inside the circle, and the other lies outside. How many separators are there?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "30",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given five points \(A, B, C, D,\) and \(E\) in the plane in general position (meaning no three of them are ever collinear, and no four of them are ever cocyclic), a separator is defined as a circle passing through three of these points such that one of the remaining two points lies inside the circle, and the other lies outside. How many separators are there?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given five points \\(A, B, C, D,\\) and \\(E\\) in the plane in general position (meaning no three of them are ever collinear, and no four of them are ever cocyclic), a separator is defined as a circle passing through three of these points such that one of the remaining two points lies inside the circle, and the other lies outside. How many separators are there? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
30
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "An ellipse has its foci at $(-4, 1)$ and $(-4, 5).$ Given that it passes through the point $(1, 3),$ its equation can be written in the form \\[\\frac{(x-h)^2}{a^2} + \\frac{(y-k)^2}{b^2} = 1\\]where $a, b, h, k$ are constants, and $a$ and $b$ are positive. Find $a+k.$\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "8",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
An ellipse has its foci at $(-4, 1)$ and $(-4, 5).$ Given that it passes through the point $(1, 3),$ its equation can be written in the form \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\]where $a, b, h, k$ are constants, and $a$ and $b$ are positive. Find $a+k.$
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "An ellipse has its foci at $(-4, 1)$ and $(-4, 5).$ Given that it passes through the point $(1, 3),$ its equation can be written in the form \\[\\frac{(x-h)^2}{a^2} + \\frac{(y-k)^2}{b^2} = 1\\]where $a, b, h, k$ are constants, and $a$ and $b$ are positive. Find $a+k.$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
8
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Suppose the graph of \\( y=g(x) \\) includes the points \\( (1,4), (2,6), \\) and \\( (3,2) \\).\n\nBased only on this information, there are two points that must be on the graph of \\( y=g(g(x)) \\). If we call these points \\( (a,b) \\) and \\( (c,d) \\), what is \\( ab + cd \\)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "20",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Suppose the graph of \( y=g(x) \) includes the points \( (1,4), (2,6), \) and \( (3,2) \).
Based only on this information, there are two points that must be on the graph of \( y=g(g(x)) \). If we call these points \( (a,b) \) and \( (c,d) \), what is \( ab + cd \)?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Suppose the graph of \\( y=g(x) \\) includes the points \\( (1,4), (2,6), \\) and \\( (3,2) \\).\n\nBased only on this information, there are two points that must be on the graph of \\( y=g(g(x)) \\). If we call these points \\( (a,b) \\) and \\( (c,d) \\), what is \\( ab + cd \\)? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
20
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "How many ways are there to put 4 distinguishable balls into 4 distinguishable boxes if one particular ball cannot be placed in one specific box?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "192",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
How many ways are there to put 4 distinguishable balls into 4 distinguishable boxes if one particular ball cannot be placed in one specific box?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "How many ways are there to put 4 distinguishable balls into 4 distinguishable boxes if one particular ball cannot be placed in one specific box? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
192
|
numina_olympiads
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Determine the number of solutions of the equation \\(\\sqrt{x+3} = ax + 2\\) depending on the value of the parameter \\(a\\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\begin{aligned}\n&\\text{Two solutions for } 0 < a < \\frac{1}{6} \\text{ and } \\frac{1}{2} < a \\leq \\frac{2}{3},\\\\\n&\\text{One solution for } a \\leq 0, a = \\frac{1}{6}, a = \\frac{1}{2}, \\text{ and } a > \\frac{2}{3},\\\\\n&\\text{No solution for } \\frac{1}{6} < a < \\frac{1}{2}.\n\\end{aligned}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Determine the number of solutions of the equation \(\sqrt{x+3} = ax + 2\) depending on the value of the parameter \(a\).
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Determine the number of solutions of the equation \\(\\sqrt{x+3} = ax + 2\\) depending on the value of the parameter \\(a\\). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\begin{aligned}
&\text{Two solutions for } 0 < a < \frac{1}{6} \text{ and } \frac{1}{2} < a \leq \frac{2}{3},\\
&\text{One solution for } a \leq 0, a = \frac{1}{6}, a = \frac{1}{2}, \text{ and } a > \frac{2}{3},\\
&\text{No solution for } \frac{1}{6} < a < \frac{1}{2}.
\end{aligned}
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given the set $U={1,2,3,4}$, set $A={1,3,4}$, and set $B={2,4}$, find the set $(∁\\_UA)∩B=$\\_\\_\\_\\_\\_\\_.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "2",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given the set $U={1,2,3,4}$, set $A={1,3,4}$, and set $B={2,4}$, find the set $(∁\_UA)∩B=$\_\_\_\_\_\_.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given the set $U={1,2,3,4}$, set $A={1,3,4}$, and set $B={2,4}$, find the set $(∁\\_UA)∩B=$\\_\\_\\_\\_\\_\\_. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
2
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given that i is the imaginary unit, find the equivalent form of the complex number $\\frac{5i}{1-2i}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "-2 + i",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given that i is the imaginary unit, find the equivalent form of the complex number $\frac{5i}{1-2i}$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given that i is the imaginary unit, find the equivalent form of the complex number $\\frac{5i}{1-2i}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
-2 + i
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "What is the value of $\\sqrt[3]{5^7 + 5^7 + 5^7}$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "3^{1/3} \\cdot 5^{7/3}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
What is the value of $\sqrt[3]{5^7 + 5^7 + 5^7}$?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "What is the value of $\\sqrt[3]{5^7 + 5^7 + 5^7}$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
3^{1/3} \cdot 5^{7/3}
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given the function $y=\\log_{a}x (a > 0, a \\neq 1)$, its graph passes through the point $(8,3)$. Determine its inverse function.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "y=2^{x}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given the function $y=\log_{a}x (a > 0, a \neq 1)$, its graph passes through the point $(8,3)$. Determine its inverse function.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given the function $y=\\log_{a}x (a > 0, a \\neq 1)$, its graph passes through the point $(8,3)$. Determine its inverse function. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
y=2^{x}
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given $\\cos\\left(\\alpha+ \\frac {\\pi}{6}\\right)= \\frac {1}{3}$, where $\\alpha\\in[0, \\frac {\\pi}{2}]$, calculate the value of $\\cos\\alpha$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac {2 \\sqrt {2}+ \\sqrt {3}}{6}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given $\cos\left(\alpha+ \frac {\pi}{6}\right)= \frac {1}{3}$, where $\alpha\in[0, \frac {\pi}{2}]$, calculate the value of $\cos\alpha$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given $\\cos\\left(\\alpha+ \\frac {\\pi}{6}\\right)= \\frac {1}{3}$, where $\\alpha\\in[0, \\frac {\\pi}{2}]$, calculate the value of $\\cos\\alpha$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac {2 \sqrt {2}+ \sqrt {3}}{6}
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Let $g(n)$ be the integer closest to $\\sqrt[5]{n}$. Find $\\sum_{k=1}^{4095}\\frac{1}{g(k)}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "2226",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Let $g(n)$ be the integer closest to $\sqrt[5]{n}$. Find $\sum_{k=1}^{4095}\frac{1}{g(k)}$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Let $g(n)$ be the integer closest to $\\sqrt[5]{n}$. Find $\\sum_{k=1}^{4095}\\frac{1}{g(k)}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
2226
|
numina_synthetic_amc
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given the equation $2x(kx - 5) - x^2 + 12 = 0$, determine the smallest integral value of $k$ for which this equation has no real roots.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "3",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given the equation $2x(kx - 5) - x^2 + 12 = 0$, determine the smallest integral value of $k$ for which this equation has no real roots.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given the equation $2x(kx - 5) - x^2 + 12 = 0$, determine the smallest integral value of $k$ for which this equation has no real roots. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
3
|
numina_olympiads
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "A sequence of natural numbers $\\{x_{n}\\}$ is constructed according to the following rules:\n$$\nx_{1}=a, \\quad x_{2}=b, \\quad x_{n+2}=x_{n}+x_{n+1}, \\quad n \\geqslant 1 .\n$$\n\nIt is known that one term in the sequence is 1000. What is the minimum possible value of $a+b$?\n\n(Note: Recommended problem by the Soviet Ministry of Education, 1990)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "10",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
A sequence of natural numbers $\{x_{n}\}$ is constructed according to the following rules:
$$
x_{1}=a, \quad x_{2}=b, \quad x_{n+2}=x_{n}+x_{n+1}, \quad n \geqslant 1 .
$$
It is known that one term in the sequence is 1000. What is the minimum possible value of $a+b$?
(Note: Recommended problem by the Soviet Ministry of Education, 1990)
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "A sequence of natural numbers $\\{x_{n}\\}$ is constructed according to the following rules:\n$$\nx_{1}=a, \\quad x_{2}=b, \\quad x_{n+2}=x_{n}+x_{n+1}, \\quad n \\geqslant 1 .\n$$\n\nIt is known that one term in the sequence is 1000. What is the minimum possible value of $a+b$?\n\n(Note: Recommended problem by the Soviet Ministry of Education, 1990) Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
10
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Calculate the units digit of the sum of $1! + 2! + 3! + \\cdots + 10!$ combined with $10^3$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "3",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Calculate the units digit of the sum of $1! + 2! + 3! + \cdots + 10!$ combined with $10^3$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Calculate the units digit of the sum of $1! + 2! + 3! + \\cdots + 10!$ combined with $10^3$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
3
|
numina_synthetic_amc
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Let $g(x)$ be a polynomial with leading coefficient $1$, whose four roots are the reciprocals of the roots of $f(x) = x^4 + ax^3 + bx^2 + cx + d$, where $a, b, c, d$ are positive integers with $a < b < c < d$. Determine the value of $g(2)$ in terms of $a, b, c,$ and $d$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac{16 + 8a + 4b + 2c + d}{pqrs}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Let $g(x)$ be a polynomial with leading coefficient $1$, whose four roots are the reciprocals of the roots of $f(x) = x^4 + ax^3 + bx^2 + cx + d$, where $a, b, c, d$ are positive integers with $a < b < c < d$. Determine the value of $g(2)$ in terms of $a, b, c,$ and $d$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Let $g(x)$ be a polynomial with leading coefficient $1$, whose four roots are the reciprocals of the roots of $f(x) = x^4 + ax^3 + bx^2 + cx + d$, where $a, b, c, d$ are positive integers with $a < b < c < d$. Determine the value of $g(2)$ in terms of $a, b, c,$ and $d$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{16 + 8a + 4b + 2c + d}{pqrs}
|
numina_olympiads
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Let $I_{n}$ be the set of the first $n$ odd natural numbers. For example: $I_{3}=\\{1,3,5\\}, I_{6}=\\{1,3,5,7,9,11\\}$, etc.\n\nFor which numbers $n$ can the set $I_{n}$ be divided into two (disjoint) parts such that the sums of the numbers in each part are equal?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "n \\text{ is even and } n \\geq 4",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Let $I_{n}$ be the set of the first $n$ odd natural numbers. For example: $I_{3}=\{1,3,5\}, I_{6}=\{1,3,5,7,9,11\}$, etc.
For which numbers $n$ can the set $I_{n}$ be divided into two (disjoint) parts such that the sums of the numbers in each part are equal?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Let $I_{n}$ be the set of the first $n$ odd natural numbers. For example: $I_{3}=\\{1,3,5\\}, I_{6}=\\{1,3,5,7,9,11\\}$, etc.\n\nFor which numbers $n$ can the set $I_{n}$ be divided into two (disjoint) parts such that the sums of the numbers in each part are equal? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
n \text{ is even and } n \geq 4
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given that $i$ is the imaginary unit, find the value of the complex number $\\dfrac{2}{1-i}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "1+i",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given that $i$ is the imaginary unit, find the value of the complex number $\dfrac{2}{1-i}$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given that $i$ is the imaginary unit, find the value of the complex number $\\dfrac{2}{1-i}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
1+i
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "The graph of the function $f(x) = \\sin(2x)$ is translated to the right by $\\frac{\\pi}{6}$ units to obtain the graph of the function $y = g(x)$. Determine the interval(s) where the function $y = g(x)$ is decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\left[\\frac{5\\pi}{12} + k\\pi, k\\pi + \\frac{11\\pi}{12}\\right] \\text{ where } k \\in \\mathbb{Z}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
The graph of the function $f(x) = \sin(2x)$ is translated to the right by $\frac{\pi}{6}$ units to obtain the graph of the function $y = g(x)$. Determine the interval(s) where the function $y = g(x)$ is decreasing.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "The graph of the function $f(x) = \\sin(2x)$ is translated to the right by $\\frac{\\pi}{6}$ units to obtain the graph of the function $y = g(x)$. Determine the interval(s) where the function $y = g(x)$ is decreasing. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\left[\frac{5\pi}{12} + k\pi, k\pi + \frac{11\pi}{12}\right] \text{ where } k \in \mathbb{Z}
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "After Liu Xiang, a famous Chinese track and field athlete, set a new world record of 12.88 seconds in the 110m hurdles, an expert group analyzed the images and data from his past competitions and training sessions. The analysis showed that the time it took him to clear each hurdle (the \"hurdle cycle\" time between two consecutive hurdles, with equal distance between them) was never more than one second, with the fastest \"hurdle cycle\" being an astonishing 0.96 seconds. The distance from the starting line to the first hurdle is 13.72 meters, and Liu Xiang's best time for this segment is 2.5 seconds; the distance from the last hurdle to the finish line is 14.02 meters, and his best time for this segment is 1.4 seconds. Based on the above data, calculate the distance between two consecutive hurdles in meters, and theoretically, the best time Liu Xiang could achieve in the 110m hurdles in seconds.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "12.54 \\, \\text{seconds}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
After Liu Xiang, a famous Chinese track and field athlete, set a new world record of 12.88 seconds in the 110m hurdles, an expert group analyzed the images and data from his past competitions and training sessions. The analysis showed that the time it took him to clear each hurdle (the "hurdle cycle" time between two consecutive hurdles, with equal distance between them) was never more than one second, with the fastest "hurdle cycle" being an astonishing 0.96 seconds. The distance from the starting line to the first hurdle is 13.72 meters, and Liu Xiang's best time for this segment is 2.5 seconds; the distance from the last hurdle to the finish line is 14.02 meters, and his best time for this segment is 1.4 seconds. Based on the above data, calculate the distance between two consecutive hurdles in meters, and theoretically, the best time Liu Xiang could achieve in the 110m hurdles in seconds.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "After Liu Xiang, a famous Chinese track and field athlete, set a new world record of 12.88 seconds in the 110m hurdles, an expert group analyzed the images and data from his past competitions and training sessions. The analysis showed that the time it took him to clear each hurdle (the \"hurdle cycle\" time between two consecutive hurdles, with equal distance between them) was never more than one second, with the fastest \"hurdle cycle\" being an astonishing 0.96 seconds. The distance from the starting line to the first hurdle is 13.72 meters, and Liu Xiang's best time for this segment is 2.5 seconds; the distance from the last hurdle to the finish line is 14.02 meters, and his best time for this segment is 1.4 seconds. Based on the above data, calculate the distance between two consecutive hurdles in meters, and theoretically, the best time Liu Xiang could achieve in the 110m hurdles in seconds. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
12.54 \, \text{seconds}
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given that $\\overrightarrow{e_{1}}$ and $\\overrightarrow{e_{2}}$ are two unit vectors with a 60° angle between them, calculate the dot product $(2\\overrightarrow{e_{1}} + \\overrightarrow{e_{2}}) \\cdot (-3\\overrightarrow{e_{1}} + 2\\overrightarrow{e_{2}}) = \\_\\_\\_\\_\\_\\_\\_.$\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "-\\frac{7}{2}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given that $\overrightarrow{e_{1}}$ and $\overrightarrow{e_{2}}$ are two unit vectors with a 60° angle between them, calculate the dot product $(2\overrightarrow{e_{1}} + \overrightarrow{e_{2}}) \cdot (-3\overrightarrow{e_{1}} + 2\overrightarrow{e_{2}}) = \_\_\_\_\_\_\_.$
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given that $\\overrightarrow{e_{1}}$ and $\\overrightarrow{e_{2}}$ are two unit vectors with a 60° angle between them, calculate the dot product $(2\\overrightarrow{e_{1}} + \\overrightarrow{e_{2}}) \\cdot (-3\\overrightarrow{e_{1}} + 2\\overrightarrow{e_{2}}) = \\_\\_\\_\\_\\_\\_\\_.$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
-\frac{7}{2}
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given that a k-base number $132_{(k)}$ is equal to the decimal number $30$, find the value of $k$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "k = 4",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given that a k-base number $132_{(k)}$ is equal to the decimal number $30$, find the value of $k$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given that a k-base number $132_{(k)}$ is equal to the decimal number $30$, find the value of $k$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
k = 4
|
numina_olympiads
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Find all integers \\( m \\) and \\( n \\) such that \\( mn \\) divides both \\( 3^m + 1 \\) and \\( 3^n + 1 \\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\{(1,1), (1,2), (2,1)\\}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Find all integers \( m \) and \( n \) such that \( mn \) divides both \( 3^m + 1 \) and \( 3^n + 1 \).
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Find all integers \\( m \\) and \\( n \\) such that \\( mn \\) divides both \\( 3^m + 1 \\) and \\( 3^n + 1 \\). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\{(1,1), (1,2), (2,1)\}
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "A customer purchased the following items at a supermarket: ① 24 bags of Nissin beef noodles at ¥1.80 per bag, with a 20% discount; ② Six packs of Master Kong iced tea at ¥1.70 per pack, with a 20% discount; ③ Five packs of Shanlin seaweed soup at ¥3.40 per pack, with no discount; ④ Three packs of Shuanghui sausage at ¥11.20 per pack, with a 10% discount. Calculate the total amount the customer needs to pay.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "¥89.96",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
A customer purchased the following items at a supermarket: ① 24 bags of Nissin beef noodles at ¥1.80 per bag, with a 20% discount; ② Six packs of Master Kong iced tea at ¥1.70 per pack, with a 20% discount; ③ Five packs of Shanlin seaweed soup at ¥3.40 per pack, with no discount; ④ Three packs of Shuanghui sausage at ¥11.20 per pack, with a 10% discount. Calculate the total amount the customer needs to pay.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "A customer purchased the following items at a supermarket: ① 24 bags of Nissin beef noodles at ¥1.80 per bag, with a 20% discount; ② Six packs of Master Kong iced tea at ¥1.70 per pack, with a 20% discount; ③ Five packs of Shanlin seaweed soup at ¥3.40 per pack, with no discount; ④ Three packs of Shuanghui sausage at ¥11.20 per pack, with a 10% discount. Calculate the total amount the customer needs to pay. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
¥89.96
|
numina_aops_forum
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Each lattice point with nonnegative coordinates is labeled with a nonnegative integer in such away that the point $(0, 0)$ is labeled by $0$ , and for every $x, y \\geq 0$ , the set of numbers labeled on thepoints $(x, y), (x, y + 1),$ and $(x + 1, y)$ is $\\{n, n + 1, n + 2\\}$ for some nonnegative integer $n$ . Determine,with proof, all possible labels for the point $(2000, 2024)$ . \n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\{n \\mid 0 \\leq n \\leq 6048, n \\equiv 0 \\pmod{3}\\}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Each lattice point with nonnegative coordinates is labeled with a nonnegative integer in such away that the point $(0, 0)$ is labeled by $0$ , and for every $x, y \geq 0$ , the set of numbers labeled on thepoints $(x, y), (x, y + 1),$ and $(x + 1, y)$ is $\{n, n + 1, n + 2\}$ for some nonnegative integer $n$ . Determine,with proof, all possible labels for the point $(2000, 2024)$ .
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Each lattice point with nonnegative coordinates is labeled with a nonnegative integer in such away that the point $(0, 0)$ is labeled by $0$ , and for every $x, y \\geq 0$ , the set of numbers labeled on thepoints $(x, y), (x, y + 1),$ and $(x + 1, y)$ is $\\{n, n + 1, n + 2\\}$ for some nonnegative integer $n$ . Determine,with proof, all possible labels for the point $(2000, 2024)$ . Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\{n \mid 0 \leq n \leq 6048, n \equiv 0 \pmod{3}\}
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given $f(x)= \\begin{cases} \\log_{a}(x+a-1),(x > 1) \\\\ (2a-1)x-a,(x\\leqslant 1)\\end{cases}$ satisfies $\\frac {f(x_{1})-f(x_{2})}{x_{1}-x_{2}} > 0$ for any real numbers $x_{1}\\neq x_{2}$, find the range of the real number $a$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "(1, 2]",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given $f(x)= \begin{cases} \log_{a}(x+a-1),(x > 1) \\ (2a-1)x-a,(x\leqslant 1)\end{cases}$ satisfies $\frac {f(x_{1})-f(x_{2})}{x_{1}-x_{2}} > 0$ for any real numbers $x_{1}\neq x_{2}$, find the range of the real number $a$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given $f(x)= \\begin{cases} \\log_{a}(x+a-1),(x > 1) \\\\ (2a-1)x-a,(x\\leqslant 1)\\end{cases}$ satisfies $\\frac {f(x_{1})-f(x_{2})}{x_{1}-x_{2}} > 0$ for any real numbers $x_{1}\\neq x_{2}$, find the range of the real number $a$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
(1, 2]
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given $$\\sin\\alpha= \\frac {4 \\sqrt {3}}{7}, \\cos(\\beta-\\alpha)= \\frac {13}{14}, \\text{ and } 0 < \\beta < \\alpha < \\frac {\\pi}{2}.$$ \n(1) Find the value of $\\tan 2\\alpha$; \n(2) Find the value of $\\cos\\beta$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac {1}{2}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given $$\sin\alpha= \frac {4 \sqrt {3}}{7}, \cos(\beta-\alpha)= \frac {13}{14}, \text{ and } 0 < \beta < \alpha < \frac {\pi}{2}.$$
(1) Find the value of $\tan 2\alpha$;
(2) Find the value of $\cos\beta$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given $$\\sin\\alpha= \\frac {4 \\sqrt {3}}{7}, \\cos(\\beta-\\alpha)= \\frac {13}{14}, \\text{ and } 0 < \\beta < \\alpha < \\frac {\\pi}{2}.$$ \n(1) Find the value of $\\tan 2\\alpha$; \n(2) Find the value of $\\cos\\beta$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac {1}{2}
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Divide 20 numbers evenly into two groups. The average of the first group is 50 with a variance of 33, and the average of the second group is 40 with a variance of 45. What is the standard deviation of the entire set of numbers?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\sigma = 8",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Divide 20 numbers evenly into two groups. The average of the first group is 50 with a variance of 33, and the average of the second group is 40 with a variance of 45. What is the standard deviation of the entire set of numbers?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Divide 20 numbers evenly into two groups. The average of the first group is 50 with a variance of 33, and the average of the second group is 40 with a variance of 45. What is the standard deviation of the entire set of numbers? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\sigma = 8
|
numina_synthetic_amc
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "A parking lot has 20 spaces in a row. Fifteen cars arrive, each of which requires one parking space, and their drivers choose spaces at random from among the available spaces. Uncle Bob then arrives in his truck, which requires 2 adjacent spaces. What is the probability that he is able to park?\nA) $\\frac{21}{25}$\nB) $\\frac{232}{323}$\nC) $\\frac{81}{140}$\nD) $\\frac{3}{5}$\nE) $\\frac{17}{28}$\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\text{B) } \\frac{232}{323}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
A parking lot has 20 spaces in a row. Fifteen cars arrive, each of which requires one parking space, and their drivers choose spaces at random from among the available spaces. Uncle Bob then arrives in his truck, which requires 2 adjacent spaces. What is the probability that he is able to park?
A) $\frac{21}{25}$
B) $\frac{232}{323}$
C) $\frac{81}{140}$
D) $\frac{3}{5}$
E) $\frac{17}{28}$
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "A parking lot has 20 spaces in a row. Fifteen cars arrive, each of which requires one parking space, and their drivers choose spaces at random from among the available spaces. Uncle Bob then arrives in his truck, which requires 2 adjacent spaces. What is the probability that he is able to park?\nA) $\\frac{21}{25}$\nB) $\\frac{232}{323}$\nC) $\\frac{81}{140}$\nD) $\\frac{3}{5}$\nE) $\\frac{17}{28}$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\text{B) } \frac{232}{323}
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Let $A=\\{x|x\\geqslant 1\\}$, $B=\\{x|x^{2}-mx\\leqslant 0\\}$. If $A\\cap B=\\{x|1\\leqslant x\\leqslant 4\\}$, then the value of $m$ is ______.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "4",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Let $A=\{x|x\geqslant 1\}$, $B=\{x|x^{2}-mx\leqslant 0\}$. If $A\cap B=\{x|1\leqslant x\leqslant 4\}$, then the value of $m$ is ______.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Let $A=\\{x|x\\geqslant 1\\}$, $B=\\{x|x^{2}-mx\\leqslant 0\\}$. If $A\\cap B=\\{x|1\\leqslant x\\leqslant 4\\}$, then the value of $m$ is ______. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
4
|
numina_olympiads
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "There is a batch of goods in a warehouse. If 3 large trucks are used, it can be completed in 4 days. If 4 small trucks are used, it can be completed in 5 days. If 20 carts are used, it can be completed in 6 days. Now, if 2 large trucks, 3 small trucks, and 7 carts are used together for 2 days, and then all are switched to carts to complete within the next 2 days, how many carts are needed each day at least in the last two days?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "15",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
There is a batch of goods in a warehouse. If 3 large trucks are used, it can be completed in 4 days. If 4 small trucks are used, it can be completed in 5 days. If 20 carts are used, it can be completed in 6 days. Now, if 2 large trucks, 3 small trucks, and 7 carts are used together for 2 days, and then all are switched to carts to complete within the next 2 days, how many carts are needed each day at least in the last two days?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "There is a batch of goods in a warehouse. If 3 large trucks are used, it can be completed in 4 days. If 4 small trucks are used, it can be completed in 5 days. If 20 carts are used, it can be completed in 6 days. Now, if 2 large trucks, 3 small trucks, and 7 carts are used together for 2 days, and then all are switched to carts to complete within the next 2 days, how many carts are needed each day at least in the last two days? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
15
|
numina_olympiads
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "The lengths of the sides of a triangle are $\\sqrt{3}, \\sqrt{4}(=2), \\sqrt{5}$. \n\nIn what ratio does the altitude perpendicular to the middle side divide it?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "1:3",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
The lengths of the sides of a triangle are $\sqrt{3}, \sqrt{4}(=2), \sqrt{5}$.
In what ratio does the altitude perpendicular to the middle side divide it?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "The lengths of the sides of a triangle are $\\sqrt{3}, \\sqrt{4}(=2), \\sqrt{5}$. \n\nIn what ratio does the altitude perpendicular to the middle side divide it? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
1:3
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "The prime factorization of 7560 is $2^3 \\times 3^3 \\times 5 \\times 7$. Find how many of its positive integer factors are perfect squares.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "4",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
The prime factorization of 7560 is $2^3 \times 3^3 \times 5 \times 7$. Find how many of its positive integer factors are perfect squares.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "The prime factorization of 7560 is $2^3 \\times 3^3 \\times 5 \\times 7$. Find how many of its positive integer factors are perfect squares. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
4
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Convert the number 87 to its equivalent in base-5. Find the first digit of the base-5 representation.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "3",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Convert the number 87 to its equivalent in base-5. Find the first digit of the base-5 representation.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Convert the number 87 to its equivalent in base-5. Find the first digit of the base-5 representation. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
3
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given that the iron barrel has a cylindrical shape with a volume of V, and the price of aluminum alloy is three times that of iron, determine the optimal ratio of the base radius r to the height h of the iron barrel to minimize the cost of the container.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac{1}{4}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given that the iron barrel has a cylindrical shape with a volume of V, and the price of aluminum alloy is three times that of iron, determine the optimal ratio of the base radius r to the height h of the iron barrel to minimize the cost of the container.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given that the iron barrel has a cylindrical shape with a volume of V, and the price of aluminum alloy is three times that of iron, determine the optimal ratio of the base radius r to the height h of the iron barrel to minimize the cost of the container. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{1}{4}
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "A license plate in a certain state consists of 5 digits, not necessarily distinct, and 3 letters, also not necessarily distinct. These eight characters may appear in any order, but the three letters must appear together in a block. How many distinct license plates are possible?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "105,\\!456,\\!000",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
A license plate in a certain state consists of 5 digits, not necessarily distinct, and 3 letters, also not necessarily distinct. These eight characters may appear in any order, but the three letters must appear together in a block. How many distinct license plates are possible?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "A license plate in a certain state consists of 5 digits, not necessarily distinct, and 3 letters, also not necessarily distinct. These eight characters may appear in any order, but the three letters must appear together in a block. How many distinct license plates are possible? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
105,\!456,\!000
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "For a positive integer $k$, $g(k)$ denotes the largest odd factor of $k$. For example, $g(3)=3$, $g(10)=5$. Let $S_{n}=g(1)+g(2)+g(3)+g(4)+\\ldots+g(2^{n})$.\n- (I) Find the values of $g(6)$ and $g(20)$;\n- (II) Calculate the values of $3S_{1}-2$, $3S_{2}-2$, $3S_{3}-2$, and from this, conjecture a general formula for $\\{S_{n}\\}$ (proof not required).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\dfrac {1}{3}(4^{n}+2)",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
For a positive integer $k$, $g(k)$ denotes the largest odd factor of $k$. For example, $g(3)=3$, $g(10)=5$. Let $S_{n}=g(1)+g(2)+g(3)+g(4)+\ldots+g(2^{n})$.
- (I) Find the values of $g(6)$ and $g(20)$;
- (II) Calculate the values of $3S_{1}-2$, $3S_{2}-2$, $3S_{3}-2$, and from this, conjecture a general formula for $\{S_{n}\}$ (proof not required).
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "For a positive integer $k$, $g(k)$ denotes the largest odd factor of $k$. For example, $g(3)=3$, $g(10)=5$. Let $S_{n}=g(1)+g(2)+g(3)+g(4)+\\ldots+g(2^{n})$.\n- (I) Find the values of $g(6)$ and $g(20)$;\n- (II) Calculate the values of $3S_{1}-2$, $3S_{2}-2$, $3S_{3}-2$, and from this, conjecture a general formula for $\\{S_{n}\\}$ (proof not required). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\dfrac {1}{3}(4^{n}+2)
|
numina_olympiads
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Find the limit of the sequence\n$$\n\\lim _{n \\rightarrow \\infty} \\frac{3 n^{5}-6 n^{4}+3 n+5}{(n+1)^{6}-(n-2)^{6}}\n$$\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac{1}{6}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Find the limit of the sequence
$$
\lim _{n \rightarrow \infty} \frac{3 n^{5}-6 n^{4}+3 n+5}{(n+1)^{6}-(n-2)^{6}}
$$
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Find the limit of the sequence\n$$\n\\lim _{n \\rightarrow \\infty} \\frac{3 n^{5}-6 n^{4}+3 n+5}{(n+1)^{6}-(n-2)^{6}}\n$$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{1}{6}
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "In how many distinct ways can I arrange my seven keys on a keychain, if I want to place my house key, car key, and office key next to each other in any order? Two arrangements are not considered different if the keys can be made to be in the same order by reflection or rotation.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "72",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
In how many distinct ways can I arrange my seven keys on a keychain, if I want to place my house key, car key, and office key next to each other in any order? Two arrangements are not considered different if the keys can be made to be in the same order by reflection or rotation.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "In how many distinct ways can I arrange my seven keys on a keychain, if I want to place my house key, car key, and office key next to each other in any order? Two arrangements are not considered different if the keys can be made to be in the same order by reflection or rotation. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
72
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "In triangle $XYZ$, where $XZ = 13$ units and $XY=12$ units, and angle $XYZ$ is a right angle at $Y$. There are two squares, $WXZY$ and $YZUV$, formed along the legs $XZ$ and $YZ$, respectively. What is the sum of the areas of the two squares?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "194 \\text{ square units}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
In triangle $XYZ$, where $XZ = 13$ units and $XY=12$ units, and angle $XYZ$ is a right angle at $Y$. There are two squares, $WXZY$ and $YZUV$, formed along the legs $XZ$ and $YZ$, respectively. What is the sum of the areas of the two squares?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "In triangle $XYZ$, where $XZ = 13$ units and $XY=12$ units, and angle $XYZ$ is a right angle at $Y$. There are two squares, $WXZY$ and $YZUV$, formed along the legs $XZ$ and $YZ$, respectively. What is the sum of the areas of the two squares? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
194 \text{ square units}
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "What is the greatest four-digit number that is two more than a multiple of 8 and four more than a multiple of 7?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "9990",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
What is the greatest four-digit number that is two more than a multiple of 8 and four more than a multiple of 7?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "What is the greatest four-digit number that is two more than a multiple of 8 and four more than a multiple of 7? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
9990
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given triangle $ABC$ with sides $a$, $b$, and $c$ opposite angles $A$, $B$, and $C$, respectively, such that $\\cos \\frac{A}{2} = \\frac{2\\sqrt{5}}{5}$ and $\\overrightarrow{AB} \\cdot \\overrightarrow{AC} = 3$.\n\n1. Find the area of triangle $ABC$.\n2. Find the minimum value of $a$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "2",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given triangle $ABC$ with sides $a$, $b$, and $c$ opposite angles $A$, $B$, and $C$, respectively, such that $\cos \frac{A}{2} = \frac{2\sqrt{5}}{5}$ and $\overrightarrow{AB} \cdot \overrightarrow{AC} = 3$.
1. Find the area of triangle $ABC$.
2. Find the minimum value of $a$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given triangle $ABC$ with sides $a$, $b$, and $c$ opposite angles $A$, $B$, and $C$, respectively, such that $\\cos \\frac{A}{2} = \\frac{2\\sqrt{5}}{5}$ and $\\overrightarrow{AB} \\cdot \\overrightarrow{AC} = 3$.\n\n1. Find the area of triangle $ABC$.\n2. Find the minimum value of $a$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
2
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given $i$ as the imaginary unit, and a complex number $z$ satisfies $\\left(1-z\\right)i=2$, calculate the value of $|\\frac{z+3}{\\overline{z}+i}|$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\sqrt{10}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given $i$ as the imaginary unit, and a complex number $z$ satisfies $\left(1-z\right)i=2$, calculate the value of $|\frac{z+3}{\overline{z}+i}|$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given $i$ as the imaginary unit, and a complex number $z$ satisfies $\\left(1-z\\right)i=2$, calculate the value of $|\\frac{z+3}{\\overline{z}+i}|$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\sqrt{10}
|
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