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": "Given that one solution to the system of linear equations $x - 2y = m$ is $\\begin{cases} x = 2 \\ y = 1 \\end{cases}$, find the value of $m$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "0",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given that one solution to the system of linear equations $x - 2y = m$ is $\begin{cases} x = 2 \ y = 1 \end{cases}$, find the value of $m$.
|
[
{
"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 that one solution to the system of linear equations $x - 2y = m$ is $\\begin{cases} x = 2 \\ y = 1 \\end{cases}$, find the value of $m$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
0
|
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": "Bully Dima placed 38 wooden toothpicks in a rectangle shape of \\(3 \\times 5\\). Then he simultaneously ignited two adjacent corners of this rectangle marked in the diagram below.\n\nIt is known that one toothpick burns in 10 seconds. How many seconds will it take for the whole structure to burn?\n\n(The fire spreads from each burned toothpick to all adjacent unburned toothpicks at a constant speed.)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "65",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Bully Dima placed 38 wooden toothpicks in a rectangle shape of \(3 \times 5\). Then he simultaneously ignited two adjacent corners of this rectangle marked in the diagram below.
It is known that one toothpick burns in 10 seconds. How many seconds will it take for the whole structure to burn?
(The fire spreads from each burned toothpick to all adjacent unburned toothpicks at a constant speed.)
|
[
{
"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": "Bully Dima placed 38 wooden toothpicks in a rectangle shape of \\(3 \\times 5\\). Then he simultaneously ignited two adjacent corners of this rectangle marked in the diagram below.\n\nIt is known that one toothpick burns in 10 seconds. How many seconds will it take for the whole structure to burn?\n\n(The fire spreads from each burned toothpick to all adjacent unburned toothpicks at a constant speed.) Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
65
|
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 sum \\( S_{n} = \\sin \\alpha \\cdot \\sin ^{2} \\frac{\\alpha}{2} + 2 \\sin \\frac{\\alpha}{2} \\cdot \\sin ^{2} \\frac{\\alpha}{4} + \\cdots + 2^{n-1} \\cdot \\sin \\frac{\\alpha}{2^{n-1}} \\cdot \\sin ^{2} \\frac{\\alpha}{2^{n}} \\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "1",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Find the sum \( S_{n} = \sin \alpha \cdot \sin ^{2} \frac{\alpha}{2} + 2 \sin \frac{\alpha}{2} \cdot \sin ^{2} \frac{\alpha}{4} + \cdots + 2^{n-1} \cdot \sin \frac{\alpha}{2^{n-1}} \cdot \sin ^{2} \frac{\alpha}{2^{n}} \).
|
[
{
"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": "Find the sum \\( S_{n} = \\sin \\alpha \\cdot \\sin ^{2} \\frac{\\alpha}{2} + 2 \\sin \\frac{\\alpha}{2} \\cdot \\sin ^{2} \\frac{\\alpha}{4} + \\cdots + 2^{n-1} \\cdot \\sin \\frac{\\alpha}{2^{n-1}} \\cdot \\sin ^{2} \\frac{\\alpha}{2^{n}} \\). 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 the sets $A=\\{1, 3, 2m+3\\}$ and $B=\\{3, m^2\\}$. If $B \\subseteq A$, then the real number $m=$ ?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "1 \\text{ or } 3",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given the sets $A=\{1, 3, 2m+3\}$ and $B=\{3, m^2\}$. If $B \subseteq A$, then the real number $m=$ ?
|
[
{
"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 the sets $A=\\{1, 3, 2m+3\\}$ and $B=\\{3, m^2\\}$. If $B \\subseteq A$, then the real number $m=$ ? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
1 \text{ or } 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 integers $1,2, \\ldots, 20$ are arranged in a circle in a certain order. For each integer $k$, Matthieu counts how many integers less than $k$ are among the 9 integers that follow $k$ in the clockwise direction; he counts this as $A(k)$. He also counts how many integers less than $k$ are among the 9 integers that follow $k$ in the counterclockwise direction; he counts this as $B(k)$. Matthieu notes that $A(k)=B(k)$ for all $k$. What is the number diametrically opposite to 11 on the circle?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "12",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
The integers $1,2, \ldots, 20$ are arranged in a circle in a certain order. For each integer $k$, Matthieu counts how many integers less than $k$ are among the 9 integers that follow $k$ in the clockwise direction; he counts this as $A(k)$. He also counts how many integers less than $k$ are among the 9 integers that follow $k$ in the counterclockwise direction; he counts this as $B(k)$. Matthieu notes that $A(k)=B(k)$ for all $k$. What is the number diametrically opposite to 11 on the circle?
|
[
{
"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 integers $1,2, \\ldots, 20$ are arranged in a circle in a certain order. For each integer $k$, Matthieu counts how many integers less than $k$ are among the 9 integers that follow $k$ in the clockwise direction; he counts this as $A(k)$. He also counts how many integers less than $k$ are among the 9 integers that follow $k$ in the counterclockwise direction; he counts this as $B(k)$. Matthieu notes that $A(k)=B(k)$ for all $k$. What is the number diametrically opposite to 11 on the circle? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
12
|
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 line $MN$ passes through the left focus $F$ of the ellipse $\\frac{x^{2}}{2}+y^{2}=1$ and intersects the ellipse at points $M$ and $N$. Line $PQ$ passes through the origin $O$ and is parallel to $MN$, intersecting the ellipse at points $P$ and $Q$. Find the value of $\\frac{|PQ|^{2}}{|MN|}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "2\\sqrt{2}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given that line $MN$ passes through the left focus $F$ of the ellipse $\frac{x^{2}}{2}+y^{2}=1$ and intersects the ellipse at points $M$ and $N$. Line $PQ$ passes through the origin $O$ and is parallel to $MN$, intersecting the ellipse at points $P$ and $Q$. Find the value of $\frac{|PQ|^{2}}{|MN|}$.
|
[
{
"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 line $MN$ passes through the left focus $F$ of the ellipse $\\frac{x^{2}}{2}+y^{2}=1$ and intersects the ellipse at points $M$ and $N$. Line $PQ$ passes through the origin $O$ and is parallel to $MN$, intersecting the ellipse at points $P$ and $Q$. Find the value of $\\frac{|PQ|^{2}}{|MN|}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
2\sqrt{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": "Triangle $ABC$ and point $P$ in the same plane are given. Point $P$ is equidistant from $A$ and $B$, angle $APB$ is twice angle $ACB$, and $\\overline{AC}$ intersects $\\overline{BP}$ at point $D$. If $PB = 5$ and $PD = 3$, calculate $AD\\cdot CD$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "14",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Triangle $ABC$ and point $P$ in the same plane are given. Point $P$ is equidistant from $A$ and $B$, angle $APB$ is twice angle $ACB$, and $\overline{AC}$ intersects $\overline{BP}$ at point $D$. If $PB = 5$ and $PD = 3$, calculate $AD\cdot CD$.
|
[
{
"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": "Triangle $ABC$ and point $P$ in the same plane are given. Point $P$ is equidistant from $A$ and $B$, angle $APB$ is twice angle $ACB$, and $\\overline{AC}$ intersects $\\overline{BP}$ at point $D$. If $PB = 5$ and $PD = 3$, calculate $AD\\cdot CD$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
14
|
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 $A=\\{x|-1<x\\leq4\\}$, and the set $B=\\{x|2<x\\leq5\\}$, then $A\\cap B=$ ?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\{x|2<x\\leq4\\}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given the set $A=\{x|-1<x\leq4\}$, and the set $B=\{x|2<x\leq5\}$, then $A\cap B=$ ?
|
[
{
"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 $A=\\{x|-1<x\\leq4\\}$, and the set $B=\\{x|2<x\\leq5\\}$, then $A\\cap B=$ ? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\{x|2<x\leq4\}
|
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 equation $x^2 - (2A)x + A0 = 0$ has positive integer solutions where $A$ is a positive single digit. How many such $A$s exist? (Since $A$ represents a digit, $A0$ represents the integer 10A.)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "A = 5",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
The equation $x^2 - (2A)x + A0 = 0$ has positive integer solutions where $A$ is a positive single digit. How many such $A$s exist? (Since $A$ represents a digit, $A0$ represents the integer 10A.)
|
[
{
"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 $x^2 - (2A)x + A0 = 0$ has positive integer solutions where $A$ is a positive single digit. How many such $A$s exist? (Since $A$ represents a digit, $A0$ represents the integer 10A.) Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
A = 5
|
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": "Determine the first term in the geometric sequence 12, 24, $a, b, 384$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac{3}{4}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Determine the first term in the geometric sequence 12, 24, $a, b, 384$.
|
[
{
"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 first term in the geometric sequence 12, 24, $a, b, 384$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{3}{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 a total of 36 students can perform the experiment, if one person can teach x classmates each time, determine the equation that represents the situation when the teacher and classmates teach each other successively in the first two classes.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "1 + x + (x+1)x = 36",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given that a total of 36 students can perform the experiment, if one person can teach x classmates each time, determine the equation that represents the situation when the teacher and classmates teach each other successively in the first two classes.
|
[
{
"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 total of 36 students can perform the experiment, if one person can teach x classmates each time, determine the equation that represents the situation when the teacher and classmates teach each other successively in the first two classes. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
1 + x + (x+1)x = 36
|
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 equation of the directrix of the parabola given by $y = 4x^2 - 3$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "y = -\\frac{19}{16}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Calculate the equation of the directrix of the parabola given by $y = 4x^2 - 3$.
|
[
{
"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 equation of the directrix of the parabola given by $y = 4x^2 - 3$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
y = -\frac{19}{16}
|
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 $\\sin \\left(x- \\frac {\\pi}{4}\\right)= \\frac {3}{5}$, calculate the value of $\\cos \\left(x+ \\frac {\\pi}{4}\\right)$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "-\\frac{3}{5}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given $\sin \left(x- \frac {\pi}{4}\right)= \frac {3}{5}$, calculate the value of $\cos \left(x+ \frac {\pi}{4}\right)$.
|
[
{
"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": "Given $\\sin \\left(x- \\frac {\\pi}{4}\\right)= \\frac {3}{5}$, calculate the value of $\\cos \\left(x+ \\frac {\\pi}{4}\\right)$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
-\frac{3}{5}
|
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": "The polynomial $(x+y)^7$ is expanded in decreasing powers of $x$. The third and fourth terms have a ratio of 2:1 \nwhen evaluated at $x=p$ and $y=q$, where $p$ and $q$ are positive numbers whose sum is three. Find the value of $p$.\nA) $\\frac{5}{13}$\nB) $\\frac{30}{13}$\nC) $\\frac{3}{7}$\nD) $\\frac{40}{13}$\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\textbf{(B)}\\ \\frac{30}{13}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
The polynomial $(x+y)^7$ is expanded in decreasing powers of $x$. The third and fourth terms have a ratio of 2:1
when evaluated at $x=p$ and $y=q$, where $p$ and $q$ are positive numbers whose sum is three. Find the value of $p$.
A) $\frac{5}{13}$
B) $\frac{30}{13}$
C) $\frac{3}{7}$
D) $\frac{40}{13}$
|
[
{
"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 polynomial $(x+y)^7$ is expanded in decreasing powers of $x$. The third and fourth terms have a ratio of 2:1 \nwhen evaluated at $x=p$ and $y=q$, where $p$ and $q$ are positive numbers whose sum is three. Find the value of $p$.\nA) $\\frac{5}{13}$\nB) $\\frac{30}{13}$\nC) $\\frac{3}{7}$\nD) $\\frac{40}{13}$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\textbf{(B)}\ \frac{30}{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": "Given point A(0, –1), and point B is on the line x - y + 1 = 0. If line AB is perpendicular to the line x + 2y - 3 = 0, then find the coordinates of point B.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "(2, 3)",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given point A(0, –1), and point B is on the line x - y + 1 = 0. If line AB is perpendicular to the line x + 2y - 3 = 0, then 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": "Given point A(0, –1), and point B is on the line x - y + 1 = 0. If line AB is perpendicular to the line x + 2y - 3 = 0, then find the coordinates of point B. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
(2, 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": "Given a street with 20 parking spots, 14 cars arrive and choose their spaces randomly, and then Uncle Bob arrives with his trailer needing 3 adjacent spaces, find the probability that Uncle Bob can find a suitable parking spot.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac{19275}{19380}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given a street with 20 parking spots, 14 cars arrive and choose their spaces randomly, and then Uncle Bob arrives with his trailer needing 3 adjacent spaces, find the probability that Uncle Bob can find a suitable parking spot.
|
[
{
"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 street with 20 parking spots, 14 cars arrive and choose their spaces randomly, and then Uncle Bob arrives with his trailer needing 3 adjacent spaces, find the probability that Uncle Bob can find a suitable parking spot. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{19275}{19380}
|
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 three roots of the equation \\[ax^3 + bx^2 + cx + d = 0\\] are $1$, $-2$, and $3$. Compute $\\frac{c}{d}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac{5}{6}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
The three roots of the equation \[ax^3 + bx^2 + cx + d = 0\] are $1$, $-2$, and $3$. Compute $\frac{c}{d}$.
|
[
{
"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": "The three roots of the equation \\[ax^3 + bx^2 + cx + d = 0\\] are $1$, $-2$, and $3$. Compute $\\frac{c}{d}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{5}{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": "The sum of the first and third of three consecutive odd integers is 130. What is the value of the second integer?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "65",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
The sum of the first and third of three consecutive odd integers is 130. What is the value of the second integer?
|
[
{
"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 sum of the first and third of three consecutive odd integers is 130. What is the value of the second integer? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
65
|
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": "$(1)$ Calculate: $(-\\frac{1}{3})^{-2}+\\sqrt[3]{-8}+|\\sqrt{3}-2|-(\\frac{1}{\\sqrt{2023}-1})^{0}+2\\sin 60^{\\circ}$.<br/>$(2)$ Factorize: $(x^{2}+4x)^{2}-(x^{2}+4x)-20$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "(x+5)(x-1)(x+2)^{2}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
$(1)$ Calculate: $(-\frac{1}{3})^{-2}+\sqrt[3]{-8}+|\sqrt{3}-2|-(\frac{1}{\sqrt{2023}-1})^{0}+2\sin 60^{\circ}$.<br/>$(2)$ Factorize: $(x^{2}+4x)^{2}-(x^{2}+4x)-20$.
|
[
{
"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": "$(1)$ Calculate: $(-\\frac{1}{3})^{-2}+\\sqrt[3]{-8}+|\\sqrt{3}-2|-(\\frac{1}{\\sqrt{2023}-1})^{0}+2\\sin 60^{\\circ}$.<br/>$(2)$ Factorize: $(x^{2}+4x)^{2}-(x^{2}+4x)-20$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
(x+5)(x-1)(x+2)^{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": "Given that $\\displaystyle\\frac{m}{n} = 25$, $\\displaystyle\\frac{p}{n} = 5$, and $\\displaystyle \\frac{p}{q} = \\frac{1}{15}$, find $\\displaystyle\\frac{m}{q}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac{1}{3}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given that $\displaystyle\frac{m}{n} = 25$, $\displaystyle\frac{p}{n} = 5$, and $\displaystyle \frac{p}{q} = \frac{1}{15}$, find $\displaystyle\frac{m}{q}$.
|
[
{
"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 $\\displaystyle\\frac{m}{n} = 25$, $\\displaystyle\\frac{p}{n} = 5$, and $\\displaystyle \\frac{p}{q} = \\frac{1}{15}$, find $\\displaystyle\\frac{m}{q}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{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": "Automobile license plates for a state consist of four letters followed by a dash and two single digits. How many different license plate combinations are possible if exactly two letters are repeated (each exactly once), and digits cannot be repeated?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "175,500",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Automobile license plates for a state consist of four letters followed by a dash and two single digits. How many different license plate combinations are possible if exactly two letters are repeated (each exactly once), and digits cannot be repeated?
|
[
{
"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": "Automobile license plates for a state consist of four letters followed by a dash and two single digits. How many different license plate combinations are possible if exactly two letters are repeated (each exactly once), and digits cannot be repeated? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
175,500
|
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": "From 3 male students and 2 female students, calculate the number of different election results in which at least one female student is elected.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "14",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
From 3 male students and 2 female students, calculate the number of different election results in which at least one female student is elected.
|
[
{
"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": "From 3 male students and 2 female students, calculate the number of different election results in which at least one female student is elected. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
14
|
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 $y=f(x)$, the equation of the tangent line at point M$(2, f(2))$ is $y=2x-8$. Find the value of $\\frac{f'(2)}{f(2)}$.\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 the function $y=f(x)$, the equation of the tangent line at point M$(2, f(2))$ is $y=2x-8$. Find the value of $\frac{f'(2)}{f(2)}$.
|
[
{
"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 the function $y=f(x)$, the equation of the tangent line at point M$(2, f(2))$ is $y=2x-8$. Find the value of $\\frac{f'(2)}{f(2)}$. 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": "Given that the volume of a sphere circumscribed around a cube is $\\frac{4\\pi}{3}$, find the edge length of this cube.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac{2\\sqrt{3}}{3}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given that the volume of a sphere circumscribed around a cube is $\frac{4\pi}{3}$, find the edge length of this cube.
|
[
{
"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 volume of a sphere circumscribed around a cube is $\\frac{4\\pi}{3}$, find the edge length of this cube. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{2\sqrt{3}}{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": "Find all two-digit numbers that are equal to three times the product of their digits.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "15 \\text{ and } 24",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Find all two-digit numbers that are equal to three times the product of their digits.
|
[
{
"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 two-digit numbers that are equal to three times the product of their digits. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
15 \text{ and } 24
|
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 table below reflects the relationship between the electricity consumption $x$ (in kilowatt-hours) and the electricity fee $y$ (in yuan) in City $M$:\n\n| Electricity Consumption (kWh) | 1 | 2 | 3 | 4 | 5 | $\\ldots$ |\n|------------------------------|-----|-----|-----|-----|-----|----------|\n| Electricity Fee (yuan) | 0.55| 1.1 | 1.65| 2.2 | 2.75| $\\ldots$ |\n\n$(1)$ Please write down the functional relationship between the electricity fee $y$ and the electricity consumption $x$ directly. \n$(2)$ If Xiao Ming's family pays an electricity fee of $40.7$ yuan in a certain month, how much electricity is consumed?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "74",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
The table below reflects the relationship between the electricity consumption $x$ (in kilowatt-hours) and the electricity fee $y$ (in yuan) in City $M$:
| Electricity Consumption (kWh) | 1 | 2 | 3 | 4 | 5 | $\ldots$ |
|------------------------------|-----|-----|-----|-----|-----|----------|
| Electricity Fee (yuan) | 0.55| 1.1 | 1.65| 2.2 | 2.75| $\ldots$ |
$(1)$ Please write down the functional relationship between the electricity fee $y$ and the electricity consumption $x$ directly.
$(2)$ If Xiao Ming's family pays an electricity fee of $40.7$ yuan in a certain month, how much electricity is consumed?
|
[
{
"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 table below reflects the relationship between the electricity consumption $x$ (in kilowatt-hours) and the electricity fee $y$ (in yuan) in City $M$:\n\n| Electricity Consumption (kWh) | 1 | 2 | 3 | 4 | 5 | $\\ldots$ |\n|------------------------------|-----|-----|-----|-----|-----|----------|\n| Electricity Fee (yuan) | 0.55| 1.1 | 1.65| 2.2 | 2.75| $\\ldots$ |\n\n$(1)$ Please write down the functional relationship between the electricity fee $y$ and the electricity consumption $x$ directly. \n$(2)$ If Xiao Ming's family pays an electricity fee of $40.7$ yuan in a certain month, how much electricity is consumed? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
74
|
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 $\\log_2 ( \\log_3 ( \\log_5 (N^2))) = 5$, determine how many different prime numbers are factors of $N$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "1",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given $\log_2 ( \log_3 ( \log_5 (N^2))) = 5$, determine how many different prime numbers are factors of $N$.
|
[
{
"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 $\\log_2 ( \\log_3 ( \\log_5 (N^2))) = 5$, determine how many different prime numbers are factors of $N$. 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 constant term in the binomial expansion of $(x- \\frac {2}{x})^{8}$ is ______.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "1120",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
The constant term in the binomial expansion of $(x- \frac {2}{x})^{8}$ 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 constant term in the binomial expansion of $(x- \\frac {2}{x})^{8}$ is ______. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
1120
|
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 ones digit of the largest power of $3$ that divides into $(18!)$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "1",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Find the ones digit of the largest power of $3$ that divides into $(18!)$.
|
[
{
"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 ones digit of the largest power of $3$ that divides into $(18!)$. 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 a geometric series \\(\\left\\{a_{n}\\right\\}\\) with the sum of its first \\(n\\) terms denoted by \\(S_{n}\\), and satisfying the equation \\(S_{n}=\\frac{\\left(a_{n}+1\\right)^{2}}{4}\\), find the value of \\(S_{20}\\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "400",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given a geometric series \(\left\{a_{n}\right\}\) with the sum of its first \(n\) terms denoted by \(S_{n}\), and satisfying the equation \(S_{n}=\frac{\left(a_{n}+1\right)^{2}}{4}\), find the value of \(S_{20}\).
|
[
{
"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 geometric series \\(\\left\\{a_{n}\\right\\}\\) with the sum of its first \\(n\\) terms denoted by \\(S_{n}\\), and satisfying the equation \\(S_{n}=\\frac{\\left(a_{n}+1\\right)^{2}}{4}\\), find the value of \\(S_{20}\\). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
400
|
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": "20 students attended a math competition. Each student who brought a pencil also brought a pen. 12 students forgot their pencils at home, and 2 students forgot their pens. How many fewer students brought a pencil compared to those who brought a pen but forgot their pencil?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "2",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
20 students attended a math competition. Each student who brought a pencil also brought a pen. 12 students forgot their pencils at home, and 2 students forgot their pens. How many fewer students brought a pencil compared to those who brought a pen but forgot their pencil?
|
[
{
"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": "20 students attended a math competition. Each student who brought a pencil also brought a pen. 12 students forgot their pencils at home, and 2 students forgot their pens. How many fewer students brought a pencil compared to those who brought a pen but forgot their pencil? 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": "The graph of the function $y = x^2$ ($x > 0$) has a tangent at the point $(a_k, a_k^2)$, which intersects the x-axis at the point with the x-coordinate $a_{k+1}$, where $k$ is a positive integer. Given $a_1 = 16$, find the sum $a_1 + a_3 + a_5$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "133",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
The graph of the function $y = x^2$ ($x > 0$) has a tangent at the point $(a_k, a_k^2)$, which intersects the x-axis at the point with the x-coordinate $a_{k+1}$, where $k$ is a positive integer. Given $a_1 = 16$, find the sum $a_1 + a_3 + a_5$.
|
[
{
"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": "The graph of the function $y = x^2$ ($x > 0$) has a tangent at the point $(a_k, a_k^2)$, which intersects the x-axis at the point with the x-coordinate $a_{k+1}$, where $k$ is a positive integer. Given $a_1 = 16$, find the sum $a_1 + a_3 + a_5$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
133
|
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 $$f(x)= \\overrightarrow {a}\\cdot \\overrightarrow {b}$$, where vectors $$\\overrightarrow {a}=( \\sqrt {3}\\sin2x, 1)$$, $$\\overrightarrow {b}=(1, \\cos2x)$$ ($x\\in\\mathbb{R}$),\n(1) Find the intervals of monotonic increase for the function $y=f(x)$;\n(2) In triangle ABC, the sides opposite angles A, B, and C are denoted as a, b, and c, respectively. Given $f(A)=2$, $a= \\sqrt {7}$, $b= \\sqrt {3}$, find the value of side length c.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "4",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given $$f(x)= \overrightarrow {a}\cdot \overrightarrow {b}$$, where vectors $$\overrightarrow {a}=( \sqrt {3}\sin2x, 1)$$, $$\overrightarrow {b}=(1, \cos2x)$$ ($x\in\mathbb{R}$),
(1) Find the intervals of monotonic increase for the function $y=f(x)$;
(2) In triangle ABC, the sides opposite angles A, B, and C are denoted as a, b, and c, respectively. Given $f(A)=2$, $a= \sqrt {7}$, $b= \sqrt {3}$, find the value of side length c.
|
[
{
"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 $$f(x)= \\overrightarrow {a}\\cdot \\overrightarrow {b}$$, where vectors $$\\overrightarrow {a}=( \\sqrt {3}\\sin2x, 1)$$, $$\\overrightarrow {b}=(1, \\cos2x)$$ ($x\\in\\mathbb{R}$),\n(1) Find the intervals of monotonic increase for the function $y=f(x)$;\n(2) In triangle ABC, the sides opposite angles A, B, and C are denoted as a, b, and c, respectively. Given $f(A)=2$, $a= \\sqrt {7}$, $b= \\sqrt {3}$, find the value of side length c. 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": "Given the function $f\\left(x\\right)=2x^{3}$, find $\\lim_{Δx→0}\\frac{f(1+Δx)-f(1)}{Δx}=\\_\\_\\_\\_\\_\\_$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "6",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given the function $f\left(x\right)=2x^{3}$, find $\lim_{Δx→0}\frac{f(1+Δx)-f(1)}{Δx}=\_\_\_\_\_\_$.
|
[
{
"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 $f\\left(x\\right)=2x^{3}$, find $\\lim_{Δx→0}\\frac{f(1+Δx)-f(1)}{Δx}=\\_\\_\\_\\_\\_\\_$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
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": "What multiple of 15 is closest to 500?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "495",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
What multiple of 15 is closest to 500?
|
[
{
"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 multiple of 15 is closest to 500? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
495
|
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, the lengths of the sides opposite to angles A, B, and C are a, b, and c, respectively. Given that sin²A + sin²C - sin²B - sinA sinC = 0.\n(1) Find the measure of angle B.\n(2) If $\\frac{a}{c} = \\frac{3}{2}$, find the value of tanC.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac{\\sqrt{3}}{2}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
In triangle ABC, the lengths of the sides opposite to angles A, B, and C are a, b, and c, respectively. Given that sin²A + sin²C - sin²B - sinA sinC = 0.
(1) Find the measure of angle B.
(2) If $\frac{a}{c} = \frac{3}{2}$, find the value of tanC.
|
[
{
"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, the lengths of the sides opposite to angles A, B, and C are a, b, and c, respectively. Given that sin²A + sin²C - sin²B - sinA sinC = 0.\n(1) Find the measure of angle B.\n(2) If $\\frac{a}{c} = \\frac{3}{2}$, find the value of tanC. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{\sqrt{3}}{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": "Seven children at a summer camp want to play pairball, a game where two children can play at a time. They have a total of 105 minutes available for the game. The children play in pairs so that each child gets an equal amount of playing time. Calculate the number of minutes each child gets to play.\nA) 15 minutes\nB) 20 minutes\nC) 25 minutes\nD) 30 minutes\nE) 35 minutes\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\text{(D)}\\ 30",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Seven children at a summer camp want to play pairball, a game where two children can play at a time. They have a total of 105 minutes available for the game. The children play in pairs so that each child gets an equal amount of playing time. Calculate the number of minutes each child gets to play.
A) 15 minutes
B) 20 minutes
C) 25 minutes
D) 30 minutes
E) 35 minutes
|
[
{
"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": "Seven children at a summer camp want to play pairball, a game where two children can play at a time. They have a total of 105 minutes available for the game. The children play in pairs so that each child gets an equal amount of playing time. Calculate the number of minutes each child gets to play.\nA) 15 minutes\nB) 20 minutes\nC) 25 minutes\nD) 30 minutes\nE) 35 minutes Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\text{(D)}\ 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": "There are 20 ordered pairs of integers $(x,y)$ that satisfy the equation $(x-1)^2 + (y+2)^2 = 36$. What is the greatest possible sum of $x+y$, and what is the lowest value of $x-y$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "-3",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
There are 20 ordered pairs of integers $(x,y)$ that satisfy the equation $(x-1)^2 + (y+2)^2 = 36$. What is the greatest possible sum of $x+y$, and what is the lowest value of $x-y$?
|
[
{
"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 are 20 ordered pairs of integers $(x,y)$ that satisfy the equation $(x-1)^2 + (y+2)^2 = 36$. What is the greatest possible sum of $x+y$, and what is the lowest value of $x-y$? 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": "Given the digits 1, 2, 3, 7, 8, 9, find the smallest sum of two 3-digit numbers that can be obtained by placing each of these digits in one of the six boxes in the given addition problem, with the condition that each number must contain one digit from 1, 2, 3 and one digit from 7, 8, 9.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "417",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given the digits 1, 2, 3, 7, 8, 9, find the smallest sum of two 3-digit numbers that can be obtained by placing each of these digits in one of the six boxes in the given addition problem, with the condition that each number must contain one digit from 1, 2, 3 and one digit from 7, 8, 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": "Given the digits 1, 2, 3, 7, 8, 9, find the smallest sum of two 3-digit numbers that can be obtained by placing each of these digits in one of the six boxes in the given addition problem, with the condition that each number must contain one digit from 1, 2, 3 and one digit from 7, 8, 9. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
417
|
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 $f(x)=2\\cos ^{2}x+\\cos (2x+ \\frac {π}{3})$.\n(1) If $f(α)= \\frac { \\sqrt {3}}{3}+1$, $0 < a < \\frac {π}{6}$, find the value of $\\sin 2α$;\n(2) In acute triangle $\\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively; if $f(A)=- \\frac {1}{2}$, $c=3$, the area of $\\triangle ABC$ is $S_{\\triangle ABC}=3 \\sqrt {3}$, find the value of $a$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\sqrt {13}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given the function $f(x)=2\cos ^{2}x+\cos (2x+ \frac {π}{3})$.
(1) If $f(α)= \frac { \sqrt {3}}{3}+1$, $0 < a < \frac {π}{6}$, find the value of $\sin 2α$;
(2) In acute triangle $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively; if $f(A)=- \frac {1}{2}$, $c=3$, the area of $\triangle ABC$ is $S_{\triangle ABC}=3 \sqrt {3}$, find the 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 the function $f(x)=2\\cos ^{2}x+\\cos (2x+ \\frac {π}{3})$.\n(1) If $f(α)= \\frac { \\sqrt {3}}{3}+1$, $0 < a < \\frac {π}{6}$, find the value of $\\sin 2α$;\n(2) In acute triangle $\\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively; if $f(A)=- \\frac {1}{2}$, $c=3$, the area of $\\triangle ABC$ is $S_{\\triangle ABC}=3 \\sqrt {3}$, find the value of $a$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\sqrt {13}
|
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": "If $y = kx^{\\frac{1}{3}}$ and $y = 4\\sqrt{3}$ at $x = 64$, what is the value of $y$ at $x = 8$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "2\\sqrt{3}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
If $y = kx^{\frac{1}{3}}$ and $y = 4\sqrt{3}$ at $x = 64$, what is the value of $y$ at $x = 8$?
|
[
{
"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 $y = kx^{\\frac{1}{3}}$ and $y = 4\\sqrt{3}$ at $x = 64$, what is the value of $y$ at $x = 8$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
2\sqrt{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": "Let \\( a \\) and \\( b \\) be real numbers such that \\( 17\\left(a^{2}+b^{2}\\right)-30ab-16=0 \\). Find the maximum value of \\( \\sqrt{16a^{2}+4b^{2}-16ab-12a+6b+9} \\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "7",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Let \( a \) and \( b \) be real numbers such that \( 17\left(a^{2}+b^{2}\right)-30ab-16=0 \). Find the maximum value of \( \sqrt{16a^{2}+4b^{2}-16ab-12a+6b+9} \).
|
[
{
"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": "Let \\( a \\) and \\( b \\) be real numbers such that \\( 17\\left(a^{2}+b^{2}\\right)-30ab-16=0 \\). Find the maximum value of \\( \\sqrt{16a^{2}+4b^{2}-16ab-12a+6b+9} \\). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
7
|
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": "Solve for $q$: $\\frac{5}{6} = \\frac{n}{60} = \\frac{m+n}{90} = \\frac{q-m}{150}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "q = 150",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Solve for $q$: $\frac{5}{6} = \frac{n}{60} = \frac{m+n}{90} = \frac{q-m}{150}$.
|
[
{
"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": "Solve for $q$: $\\frac{5}{6} = \\frac{n}{60} = \\frac{m+n}{90} = \\frac{q-m}{150}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
q = 150
|
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": "For how many non-negative real values of $x$ is $\\sqrt{169-\\sqrt[4]{x}}$ an integer?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "14",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
For how many non-negative real values of $x$ is $\sqrt{169-\sqrt[4]{x}}$ an integer?
|
[
{
"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 how many non-negative real values of $x$ is $\\sqrt{169-\\sqrt[4]{x}}$ an integer? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
14
|
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 TV station is broadcasting 5 advertisements in a row, including 3 different commercial advertisements and 2 different public service advertisements. The last advertisement cannot be a commercial one, and the two public service advertisements cannot be broadcast consecutively. How many different broadcasting methods are there? (Answer with a number).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "36",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
A TV station is broadcasting 5 advertisements in a row, including 3 different commercial advertisements and 2 different public service advertisements. The last advertisement cannot be a commercial one, and the two public service advertisements cannot be broadcast consecutively. How many different broadcasting methods are there? (Answer with a 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": "A TV station is broadcasting 5 advertisements in a row, including 3 different commercial advertisements and 2 different public service advertisements. The last advertisement cannot be a commercial one, and the two public service advertisements cannot be broadcast consecutively. How many different broadcasting methods are there? (Answer with a number). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
36
|
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 the 15th term of the arithmetic sequence $3, 6, 9, \\ldots$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "45",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
What is the value of the 15th term of the arithmetic sequence $3, 6, 9, \ldots$?
|
[
{
"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 the 15th term of the arithmetic sequence $3, 6, 9, \\ldots$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
45
|
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": "For how many integers \\(n\\) between 1 and 60, inclusive, is \\[\\frac{(n^2-1)!}{(n!)^n}\\] an integer?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "42",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
For how many integers \(n\) between 1 and 60, inclusive, is \[\frac{(n^2-1)!}{(n!)^n}\] an integer?
|
[
{
"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 how many integers \\(n\\) between 1 and 60, inclusive, is \\[\\frac{(n^2-1)!}{(n!)^n}\\] an integer? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
42
|
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 a right-angled triangle $PQR$ with angle $PQR = 90^{\\circ}$, suppose $\\cos Q = \\frac{5}{13}$. Given $PR = 13$, calculate the length of $PQ$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "12",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
In a right-angled triangle $PQR$ with angle $PQR = 90^{\circ}$, suppose $\cos Q = \frac{5}{13}$. Given $PR = 13$, calculate the length of $PQ$.
|
[
{
"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 a right-angled triangle $PQR$ with angle $PQR = 90^{\\circ}$, suppose $\\cos Q = \\frac{5}{13}$. Given $PR = 13$, calculate the length of $PQ$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
12
|
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": "Simplify $\\dfrac{\\cos (\\dfrac{5\\pi }{2}-\\alpha )\\cdot \\cos (-\\alpha )}{\\sin (\\dfrac{3\\pi }{2}+\\alpha )\\cdot \\cos (\\dfrac{21\\pi }{2}-\\alpha )}$ = \\_\\_\\_ \\_\\_\\_\\_．\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "-1",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Simplify $\dfrac{\cos (\dfrac{5\pi }{2}-\alpha )\cdot \cos (-\alpha )}{\sin (\dfrac{3\pi }{2}+\alpha )\cdot \cos (\dfrac{21\pi }{2}-\alpha )}$ = \_\_\_ \_\_\_\_．
|
[
{
"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": "Simplify $\\dfrac{\\cos (\\dfrac{5\\pi }{2}-\\alpha )\\cdot \\cos (-\\alpha )}{\\sin (\\dfrac{3\\pi }{2}+\\alpha )\\cdot \\cos (\\dfrac{21\\pi }{2}-\\alpha )}$ = \\_\\_\\_ \\_\\_\\_\\_． 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": "Find all composite positive integers \\( n \\) such that all the divisors of \\( n \\) can be written in the form \\( a^{r} + 1 \\), where \\( a \\) and \\( r \\) are integers with \\( a \\geq 0 \\) and \\( r \\geq 2 \\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "10",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Find all composite positive integers \( n \) such that all the divisors of \( n \) can be written in the form \( a^{r} + 1 \), where \( a \) and \( r \) are integers with \( a \geq 0 \) and \( r \geq 2 \).
|
[
{
"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": "Find all composite positive integers \\( n \\) such that all the divisors of \\( n \\) can be written in the form \\( a^{r} + 1 \\), where \\( a \\) and \\( r \\) are integers with \\( a \\geq 0 \\) and \\( r \\geq 2 \\). 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": "Determine the order from greatest to least of the following values: $\\sin(1)$, $\\sin(2)$, $\\sin(3)$, $\\sin(4)$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\sin(2) > \\sin(1) > \\sin(3) > \\sin(4)",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Determine the order from greatest to least of the following values: $\sin(1)$, $\sin(2)$, $\sin(3)$, $\sin(4)$.
|
[
{
"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": "Determine the order from greatest to least of the following values: $\\sin(1)$, $\\sin(2)$, $\\sin(3)$, $\\sin(4)$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\sin(2) > \sin(1) > \sin(3) > \sin(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": "Consider a matrix $A = \\begin{pmatrix} a_{11} & a_{12} \\\\ a_{21} & a_{22} \\end{pmatrix}$ where $a_{11}, a_{12}, a_{21}, a_{22} \\in \\{0, 1\\}$, and the determinant of $A$ is 0. Determine the number of distinct matrices $A$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "10",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Consider a matrix $A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}$ where $a_{11}, a_{12}, a_{21}, a_{22} \in \{0, 1\}$, and the determinant of $A$ is 0. Determine the number of distinct matrices $A$.
|
[
{
"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": "Consider a matrix $A = \\begin{pmatrix} a_{11} & a_{12} \\\\ a_{21} & a_{22} \\end{pmatrix}$ where $a_{11}, a_{12}, a_{21}, a_{22} \\in \\{0, 1\\}$, and the determinant of $A$ is 0. Determine the number of distinct matrices $A$. 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": "Determine the range of the function $y = x + \\frac{4}{x}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "y \\leq -4 \\quad \\text{or} \\quad y \\geq 4",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Determine the range of the function $y = x + \frac{4}{x}$.
|
[
{
"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": "Determine the range of the function $y = x + \\frac{4}{x}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
y \leq -4 \quad \text{or} \quad y \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 the sets $M=\\{(x,y)|y-1=k(x+1), x, y \\in \\mathbb{R}\\}$ and $N=\\{(x,y)|x^2+y^2-2y=0, x, y \\in \\mathbb{R}\\}$, find the number of elements in the intersection of M and N.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "1",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given the sets $M=\{(x,y)|y-1=k(x+1), x, y \in \mathbb{R}\}$ and $N=\{(x,y)|x^2+y^2-2y=0, x, y \in \mathbb{R}\}$, find the number of elements in the intersection of 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 sets $M=\\{(x,y)|y-1=k(x+1), x, y \\in \\mathbb{R}\\}$ and $N=\\{(x,y)|x^2+y^2-2y=0, x, y \\in \\mathbb{R}\\}$, find the number of elements in the intersection of M and N. 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 an ellipse $C$: $\\dfrac{x^2}{a^2} + \\dfrac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $\\dfrac{1}{2}$, and a line $l$: $x + 2y = 4$ that intersects the ellipse at exactly one point $T$.\n\n$(I)$ Find the equation of ellipse $C$ and the coordinates of point $T$; \n$(II)$ Let $O$ be the origin, and a line $l'$ parallel to $OT$ intersects the ellipse $C$ at two distinct points $A$ and $B$. Find the equation of line $l'$ when the area of $\\triangle OAB$ is maximized.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "y = \\dfrac{3}{2}x - \\sqrt{6}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given an ellipse $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $\dfrac{1}{2}$, and a line $l$: $x + 2y = 4$ that intersects the ellipse at exactly one point $T$.
$(I)$ Find the equation of ellipse $C$ and the coordinates of point $T$;
$(II)$ Let $O$ be the origin, and a line $l'$ parallel to $OT$ intersects the ellipse $C$ at two distinct points $A$ and $B$. Find the equation of line $l'$ when the area of $\triangle OAB$ is maximized.
|
[
{
"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 ellipse $C$: $\\dfrac{x^2}{a^2} + \\dfrac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $\\dfrac{1}{2}$, and a line $l$: $x + 2y = 4$ that intersects the ellipse at exactly one point $T$.\n\n$(I)$ Find the equation of ellipse $C$ and the coordinates of point $T$; \n$(II)$ Let $O$ be the origin, and a line $l'$ parallel to $OT$ intersects the ellipse $C$ at two distinct points $A$ and $B$. Find the equation of line $l'$ when the area of $\\triangle OAB$ is maximized. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
y = \dfrac{3}{2}x - \sqrt{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": "Two different points, $E$ and $F$, lie on opposite sides of line $AB$ such that triangles $ABE$ and $ABF$ are congruent with $AB = 12$, $BE = AF = 13$, and $EA = FB = 20$. Another point $G$ lies on line segment $AB$ such that $AG = 5$ and $GB = 7$. Determine the area of intersection of triangles $ABE$ and $ABG$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "71.28",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Two different points, $E$ and $F$, lie on opposite sides of line $AB$ such that triangles $ABE$ and $ABF$ are congruent with $AB = 12$, $BE = AF = 13$, and $EA = FB = 20$. Another point $G$ lies on line segment $AB$ such that $AG = 5$ and $GB = 7$. Determine the area of intersection of triangles $ABE$ and $ABG$.
|
[
{
"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": "Two different points, $E$ and $F$, lie on opposite sides of line $AB$ such that triangles $ABE$ and $ABF$ are congruent with $AB = 12$, $BE = AF = 13$, and $EA = FB = 20$. Another point $G$ lies on line segment $AB$ such that $AG = 5$ and $GB = 7$. Determine the area of intersection of triangles $ABE$ and $ABG$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
71.28
|
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 system of equations $\\left\\{\\begin{array}{l}{x+y=4}\\\\{y+z=6}\\\\{z+x=8}\\end{array}\\right.$, determine the value of $x+y+z$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "9",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given the system of equations $\left\{\begin{array}{l}{x+y=4}\\{y+z=6}\\{z+x=8}\end{array}\right.$, determine the value of $x+y+z$.
|
[
{
"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 the system of equations $\\left\\{\\begin{array}{l}{x+y=4}\\\\{y+z=6}\\\\{z+x=8}\\end{array}\\right.$, determine the value of $x+y+z$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
9
|
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": "Place 6 cards labeled 1, 2, 3, 4, 5, 6 into 3 different envelopes, with each envelope containing 2 cards. If the cards labeled 1 and 2 are placed in the same envelope, calculate the total number of different methods.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "18",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Place 6 cards labeled 1, 2, 3, 4, 5, 6 into 3 different envelopes, with each envelope containing 2 cards. If the cards labeled 1 and 2 are placed in the same envelope, calculate the total number of different methods.
|
[
{
"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": "Place 6 cards labeled 1, 2, 3, 4, 5, 6 into 3 different envelopes, with each envelope containing 2 cards. If the cards labeled 1 and 2 are placed in the same envelope, calculate the total number of different methods. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
18
|
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 $a\\_1$ and $a\\__{2017}$ are the two roots of the equation $x^{2}-10x+16=0$ in the arithmetic sequence $\\{a\\_n\\}$, find the value of $a\\_2 + a\\__{1009} + a\\__{2016}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "15",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given that $a\_1$ and $a\__{2017}$ are the two roots of the equation $x^{2}-10x+16=0$ in the arithmetic sequence $\{a\_n\}$, find the value of $a\_2 + a\__{1009} + a\__{2016}$.
|
[
{
"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 $a\\_1$ and $a\\__{2017}$ are the two roots of the equation $x^{2}-10x+16=0$ in the arithmetic sequence $\\{a\\_n\\}$, find the value of $a\\_2 + a\\__{1009} + a\\__{2016}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
15
|
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": "Ike and Mike go into a sandwich shop with a total of $35.00 to spend. Sandwiches cost $5.00 each and soft drinks cost $1.50 each. Ike and Mike plan to buy as many sandwiches as they can, and use any remaining money to buy soft drinks. Counting both sandwiches and soft drinks, how many items will they buy?\nA) 6 \nB) 7 \nC) 8 \nD) 9 \nE) 10\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\textbf{(B) }7",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Ike and Mike go into a sandwich shop with a total of $35.00 to spend. Sandwiches cost $5.00 each and soft drinks cost $1.50 each. Ike and Mike plan to buy as many sandwiches as they can, and use any remaining money to buy soft drinks. Counting both sandwiches and soft drinks, how many items will they buy?
A) 6
B) 7
C) 8
D) 9
E) 10
|
[
{
"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": "Ike and Mike go into a sandwich shop with a total of $35.00 to spend. Sandwiches cost $5.00 each and soft drinks cost $1.50 each. Ike and Mike plan to buy as many sandwiches as they can, and use any remaining money to buy soft drinks. Counting both sandwiches and soft drinks, how many items will they buy?\nA) 6 \nB) 7 \nC) 8 \nD) 9 \nE) 10 Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\textbf{(B) }7
|
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": "Determine the length of side QR in an isosceles triangle PQRS with PQ = PR = 5 inches and the median PM from P to the midpoint M of side QR equal to 4 inches.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "6",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Determine the length of side QR in an isosceles triangle PQRS with PQ = PR = 5 inches and the median PM from P to the midpoint M of side QR equal to 4 inches.
|
[
{
"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 length of side QR in an isosceles triangle PQRS with PQ = PR = 5 inches and the median PM from P to the midpoint M of side QR equal to 4 inches. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
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": "How many three-digit numbers are composed of three distinct digits such that one digit is twice the difference between the other two?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "90",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
How many three-digit numbers are composed of three distinct digits such that one digit is twice the difference between the other two?
|
[
{
"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 three-digit numbers are composed of three distinct digits such that one digit is twice the difference between the other two? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
90
|
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 equation $2^{2x}+(m^{2}-2m-5)2^{x}+1=0$ has a solution for $x$, then the range of $m$ is ______.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "[-1,3]",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
If the equation $2^{2x}+(m^{2}-2m-5)2^{x}+1=0$ has a solution for $x$, then the range 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": "If the equation $2^{2x}+(m^{2}-2m-5)2^{x}+1=0$ has a solution for $x$, then the range of $m$ is ______. 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": "Calculate the value of $\\dfrac{11!}{7!4!} \\times 2$ without using a calculator.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "660",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Calculate the value of $\dfrac{11!}{7!4!} \times 2$ without using a calculator.
|
[
{
"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 $\\dfrac{11!}{7!4!} \\times 2$ without using a calculator. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
660
|
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": "Find the slope of the tangent line to the function $y=\\frac{2}{3}\\sqrt{{x^3}}+1$ at $x=3$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac{\\pi}{3}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Find the slope of the tangent line to the function $y=\frac{2}{3}\sqrt{{x^3}}+1$ at $x=3$.
|
[
{
"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": "Find the slope of the tangent line to the function $y=\\frac{2}{3}\\sqrt{{x^3}}+1$ at $x=3$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{\pi}{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 proposition $p: a=-1$, proposition $q$: complex number $z=\\frac{1+i}{1+ai}$, where $a\\in R$, is purely imaginary, determine the relationship between proposition $p$ and proposition $q$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "-1",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given proposition $p: a=-1$, proposition $q$: complex number $z=\frac{1+i}{1+ai}$, where $a\in R$, is purely imaginary, determine the relationship between proposition $p$ and proposition $q$.
|
[
{
"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 proposition $p: a=-1$, proposition $q$: complex number $z=\\frac{1+i}{1+ai}$, where $a\\in R$, is purely imaginary, determine the relationship between proposition $p$ and proposition $q$. 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": "Calculate the derivatives of the following functions:\n(1) $y=2e^x$;\n(2) $y=2x^5-3x^2+5x-4$;\n(3) $y=3\\cos x - 4\\sin x$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "-3\\sin x - 4\\cos x",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Calculate the derivatives of the following functions:
(1) $y=2e^x$;
(2) $y=2x^5-3x^2+5x-4$;
(3) $y=3\cos x - 4\sin x$.
|
[
{
"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": "Calculate the derivatives of the following functions:\n(1) $y=2e^x$;\n(2) $y=2x^5-3x^2+5x-4$;\n(3) $y=3\\cos x - 4\\sin x$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
-3\sin x - 4\cos 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": "A basketball player has the same free-throw shooting percentage for every attempt, and the probability that they make at most one basket in two attempts is $\\frac{16}{25}$. Find the player's free-throw shooting percentage for each attempt.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac{3}{5}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
A basketball player has the same free-throw shooting percentage for every attempt, and the probability that they make at most one basket in two attempts is $\frac{16}{25}$. Find the player's free-throw shooting percentage for each attempt.
|
[
{
"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 basketball player has the same free-throw shooting percentage for every attempt, and the probability that they make at most one basket in two attempts is $\\frac{16}{25}$. Find the player's free-throw shooting percentage for each attempt. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{3}{5}
|
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 $R$ be the set of points $(x, y)$ in the Cartesian plane that satisfy:\n\\[\n\\Big|\\big| |x|-3\\big|-2\\Big|+\\Big|\\big| |y|-3\\big|-2\\Big|=2.\n\\]\nDetermine the total length of all the lines that make up $R$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "32\\sqrt{2}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Let $R$ be the set of points $(x, y)$ in the Cartesian plane that satisfy:
\[
\Big|\big| |x|-3\big|-2\Big|+\Big|\big| |y|-3\big|-2\Big|=2.
\]
Determine the total length of all the lines that make up $R$.
|
[
{
"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 $R$ be the set of points $(x, y)$ in the Cartesian plane that satisfy:\n\\[\n\\Big|\\big| |x|-3\\big|-2\\Big|+\\Big|\\big| |y|-3\\big|-2\\Big|=2.\n\\]\nDetermine the total length of all the lines that make up $R$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
32\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 that the random variable $X$ follows a binomial distribution $X \\sim B(6, \\frac{1}{4})$, find the value of $EX$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac{3}{2}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given that the random variable $X$ follows a binomial distribution $X \sim B(6, \frac{1}{4})$, find the value of $EX$.
|
[
{
"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 random variable $X$ follows a binomial distribution $X \\sim B(6, \\frac{1}{4})$, find the value of $EX$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{3}{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": "\\((x_1, x_2, \\ldots, x_n)\\) is a permutation of \\((1, 2, \\ldots, n)\\). What is the maximum of \\(x_1 x_2 + x_2 x_3 + \\ldots + x_{n-1} x_n + x_n x_1\\)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac{2n^3 + 3n^2 - 11n + 18}{6}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
\((x_1, x_2, \ldots, x_n)\) is a permutation of \((1, 2, \ldots, n)\). What is the maximum of \(x_1 x_2 + x_2 x_3 + \ldots + x_{n-1} x_n + x_n x_1\)?
|
[
{
"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": "\\((x_1, x_2, \\ldots, x_n)\\) is a permutation of \\((1, 2, \\ldots, n)\\). What is the maximum of \\(x_1 x_2 + x_2 x_3 + \\ldots + x_{n-1} x_n + x_n x_1\\)? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{2n^3 + 3n^2 - 11n + 18}{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": "A club has 12 members, 7 boys and 5 girls. Three of the members are chosen at random. What is the probability that they are all girls?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac{1}{22}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
A club has 12 members, 7 boys and 5 girls. Three of the members are chosen at random. What is the probability that they are all girls?
|
[
{
"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 club has 12 members, 7 boys and 5 girls. Three of the members are chosen at random. What is the probability that they are all girls? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{1}{22}
|
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 right circular cone with a base radius $r$ and height $h$ lies on its side on a flat table. As the cone rolls on the surface of the table without slipping, the point where the cone's base connects with the table traces a circular arc centered at the vertex of the cone. The cone first returns to its original position after making $20$ complete rotations. The value of $h/r$ in simplest form can be expressed as $\\lambda\\sqrt{k}$, where $\\lambda$ and $k$ are positive integers, and $k$ is not divisible by the square of any prime. Find $\\lambda + k$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "400",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
A right circular cone with a base radius $r$ and height $h$ lies on its side on a flat table. As the cone rolls on the surface of the table without slipping, the point where the cone's base connects with the table traces a circular arc centered at the vertex of the cone. The cone first returns to its original position after making $20$ complete rotations. The value of $h/r$ in simplest form can be expressed as $\lambda\sqrt{k}$, where $\lambda$ and $k$ are positive integers, and $k$ is not divisible by the square of any prime. Find $\lambda + k$.
|
[
{
"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": "A right circular cone with a base radius $r$ and height $h$ lies on its side on a flat table. As the cone rolls on the surface of the table without slipping, the point where the cone's base connects with the table traces a circular arc centered at the vertex of the cone. The cone first returns to its original position after making $20$ complete rotations. The value of $h/r$ in simplest form can be expressed as $\\lambda\\sqrt{k}$, where $\\lambda$ and $k$ are positive integers, and $k$ is not divisible by the square of any prime. Find $\\lambda + k$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
400
|
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 sets $A=\\{-1,0,1\\}$ and $B=\\{a,a^{2}\\}$, find the value(s) of $a$ that make $A\\cup B=A$ true.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "a=-1",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given the sets $A=\{-1,0,1\}$ and $B=\{a,a^{2}\}$, find the value(s) of $a$ that make $A\cup B=A$ true.
|
[
{
"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 sets $A=\\{-1,0,1\\}$ and $B=\\{a,a^{2}\\}$, find the value(s) of $a$ that make $A\\cup B=A$ true. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
a=-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": "A convex polyhedron has dihedral angles at each edge that are acute. How many faces can the polyhedron have?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "4",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
A convex polyhedron has dihedral angles at each edge that are acute. How many faces can the polyhedron have?
|
[
{
"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 convex polyhedron has dihedral angles at each edge that are acute. How many faces can the polyhedron have? 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": "Define a function $f$ recursively by $f(1) = 1$, $f(2) = 2$, and\n\\[ f(n) = 2f(n-1) - f(n-2) + n^2 \\]\nfor all integers $n \\geq 3$. What is $f(10)$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "358",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Define a function $f$ recursively by $f(1) = 1$, $f(2) = 2$, and
\[ f(n) = 2f(n-1) - f(n-2) + n^2 \]
for all integers $n \geq 3$. What is $f(10)$?
|
[
{
"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": "Define a function $f$ recursively by $f(1) = 1$, $f(2) = 2$, and\n\\[ f(n) = 2f(n-1) - f(n-2) + n^2 \\]\nfor all integers $n \\geq 3$. What is $f(10)$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
358
|
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 that the complex numbers \\( z_1 \\) and \\( z_2 \\) satisfy \\( \\left|z_1\\right| < 1 \\), \\( \\left|z_2\\right| < 1 \\), \\( z_1 + z_2 \\) is real, and \\( z_1 + z_2 + z_1 z_2 = 0 \\), find the range of values of \\( z_1 z_2 \\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "z_{1} z_{2} \\in \\left( -\\frac{1}{2}, 1 \\right)",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given that the complex numbers \( z_1 \) and \( z_2 \) satisfy \( \left|z_1\right| < 1 \), \( \left|z_2\right| < 1 \), \( z_1 + z_2 \) is real, and \( z_1 + z_2 + z_1 z_2 = 0 \), find the range of values of \( z_1 z_2 \).
|
[
{
"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": "Given that the complex numbers \\( z_1 \\) and \\( z_2 \\) satisfy \\( \\left|z_1\\right| < 1 \\), \\( \\left|z_2\\right| < 1 \\), \\( z_1 + z_2 \\) is real, and \\( z_1 + z_2 + z_1 z_2 = 0 \\), find the range of values of \\( z_1 z_2 \\). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
z_{1} z_{2} \in \left( -\frac{1}{2}, 1 \right)
|
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 unit vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ satisfy $(2\\overrightarrow{a}+\\overrightarrow{b})\\bot \\overrightarrow{b}$, calculate the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac{2\\pi}{3}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $(2\overrightarrow{a}+\overrightarrow{b})\bot \overrightarrow{b}$, calculate the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$.
|
[
{
"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": "Given unit vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ satisfy $(2\\overrightarrow{a}+\\overrightarrow{b})\\bot \\overrightarrow{b}$, calculate the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{2\pi}{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": "In the polar coordinate system, the pole is O, and the polar coordinates of point A are $(2, \\frac{\\pi}{6})$. An isosceles right triangle OAB is formed with OA as the hypotenuse (where O, A, B are distributed in a counterclockwise direction) \n(1) Find the polar coordinates of point B; \n(2) Find the polar equation of the circumscribed circle of the triangle.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\rho = 2\\sin(\\theta + \\frac{\\pi}{3})",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
In the polar coordinate system, the pole is O, and the polar coordinates of point A are $(2, \frac{\pi}{6})$. An isosceles right triangle OAB is formed with OA as the hypotenuse (where O, A, B are distributed in a counterclockwise direction)
(1) Find the polar coordinates of point B;
(2) Find the polar equation of the circumscribed circle of the triangle.
|
[
{
"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 polar coordinate system, the pole is O, and the polar coordinates of point A are $(2, \\frac{\\pi}{6})$. An isosceles right triangle OAB is formed with OA as the hypotenuse (where O, A, B are distributed in a counterclockwise direction) \n(1) Find the polar coordinates of point B; \n(2) Find the polar equation of the circumscribed circle of the triangle. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\rho = 2\sin(\theta + \frac{\pi}{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": "In a $4 \\times 4$ grid, each cell is filled with the number 1 or 2. Each $3 \\times 3$ sub-grid must have the sum of its 9 numbers divisible by 4, and the sum of all 16 numbers in the grid must not be divisible by 4. What is the maximum possible sum of these 16 numbers? What is the minimum possible sum of these 16 numbers?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "19",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
In a $4 \times 4$ grid, each cell is filled with the number 1 or 2. Each $3 \times 3$ sub-grid must have the sum of its 9 numbers divisible by 4, and the sum of all 16 numbers in the grid must not be divisible by 4. What is the maximum possible sum of these 16 numbers? What is the minimum possible sum of these 16 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": "In a $4 \\times 4$ grid, each cell is filled with the number 1 or 2. Each $3 \\times 3$ sub-grid must have the sum of its 9 numbers divisible by 4, and the sum of all 16 numbers in the grid must not be divisible by 4. What is the maximum possible sum of these 16 numbers? What is the minimum possible sum of these 16 numbers? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
19
|
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": "Through the end of a chord that divides the circle in the ratio 3:5, a tangent is drawn. Find the acute angle between the chord and the tangent.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "67^\\circ 30'",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Through the end of a chord that divides the circle in the ratio 3:5, a tangent is drawn. Find the acute angle between the chord and the tangent.
|
[
{
"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": "Through the end of a chord that divides the circle in the ratio 3:5, a tangent is drawn. Find the acute angle between the chord and the tangent. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
67^\circ 30'
|
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 are 1996 points uniformly distributed on the circumference of a circle. If four distinct points \\( A \\), \\( B \\), \\( C \\), and \\( D \\) are selected from these points, calculate the probability that the chords \\( AB \\) and \\( CD \\) intersect.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac{1}{4}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
There are 1996 points uniformly distributed on the circumference of a circle. If four distinct points \( A \), \( B \), \( C \), and \( D \) are selected from these points, calculate the probability that the chords \( AB \) and \( CD \) intersect.
|
[
{
"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 are 1996 points uniformly distributed on the circumference of a circle. If four distinct points \\( A \\), \\( B \\), \\( C \\), and \\( D \\) are selected from these points, calculate the probability that the chords \\( AB \\) and \\( CD \\) intersect. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{1}{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": "Segment $CB$ is obtained by translating segment $AD$. The corresponding point of point $A(-1,4)$ is $C(3,6)$. The coordinates of the corresponding point of $D(-3,2)$ can be found by applying the same translation.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "(1,4)",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Segment $CB$ is obtained by translating segment $AD$. The corresponding point of point $A(-1,4)$ is $C(3,6)$. The coordinates of the corresponding point of $D(-3,2)$ can be found by applying the same translation.
|
[
{
"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": "Segment $CB$ is obtained by translating segment $AD$. The corresponding point of point $A(-1,4)$ is $C(3,6)$. The coordinates of the corresponding point of $D(-3,2)$ can be found by applying the same translation. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
(1,4)
|
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": "Which of the following numbers has the largest prime factor?\n- 46\n- 69\n- 87\n- 121\n- 143\nA) 46\nB) 69\nC) 87\nD) 121\nE) 143\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "The correct answer is C) 87.",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Which of the following numbers has the largest prime factor?
- 46
- 69
- 87
- 121
- 143
A) 46
B) 69
C) 87
D) 121
E) 143
|
[
{
"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": "Which of the following numbers has the largest prime factor?\n- 46\n- 69\n- 87\n- 121\n- 143\nA) 46\nB) 69\nC) 87\nD) 121\nE) 143 Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
The correct answer is C) 87.
|
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 positive difference between the probability of a fair coin landing heads up exactly 4 times out of 5 flips and the probability of a fair coin landing heads up 5 times out of 5 flips? Express your answer as a common fraction.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac{9}{32}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
What is the positive difference between the probability of a fair coin landing heads up exactly 4 times out of 5 flips and the probability of a fair coin landing heads up 5 times out of 5 flips? Express your answer as a common fraction.
|
[
{
"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 positive difference between the probability of a fair coin landing heads up exactly 4 times out of 5 flips and the probability of a fair coin landing heads up 5 times out of 5 flips? Express your answer as a common fraction. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{9}{32}
|
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 that Chloe's telephone numbers have the form $555-ab-cdef$, where $a$, $b$, $c$, $d$, $e$, and $f$ are distinct digits, in descending order, and are chosen between $1$ and $8$, calculate the total number of possible telephone numbers that Chloe can have.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "28",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given that Chloe's telephone numbers have the form $555-ab-cdef$, where $a$, $b$, $c$, $d$, $e$, and $f$ are distinct digits, in descending order, and are chosen between $1$ and $8$, calculate the total number of possible telephone numbers that Chloe can have.
|
[
{
"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 Chloe's telephone numbers have the form $555-ab-cdef$, where $a$, $b$, $c$, $d$, $e$, and $f$ are distinct digits, in descending order, and are chosen between $1$ and $8$, calculate the total number of possible telephone numbers that Chloe can have. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
28
|
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 vectors $\\overrightarrow{a} = (-1, 3)$ and $\\overrightarrow{b} = (2, y)$, if $\\overrightarrow{a} \\parallel \\overrightarrow{b}$, then the value of the real number $y$ is ______.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "-6",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given vectors $\overrightarrow{a} = (-1, 3)$ and $\overrightarrow{b} = (2, y)$, if $\overrightarrow{a} \parallel \overrightarrow{b}$, then the value of the real number $y$ 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": "Given vectors $\\overrightarrow{a} = (-1, 3)$ and $\\overrightarrow{b} = (2, y)$, if $\\overrightarrow{a} \\parallel \\overrightarrow{b}$, then the value of the real number $y$ is ______. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
-6
|
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 in the Cartesian coordinate system, there is a point A(0,1), and vectors $\\overrightarrow{AB}=(-4,-3)$, $\\overrightarrow{BC}=(-7,-4)$. Calculate the coordinates of point C.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "(-11,-6)",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given in the Cartesian coordinate system, there is a point A(0,1), and vectors $\overrightarrow{AB}=(-4,-3)$, $\overrightarrow{BC}=(-7,-4)$. Calculate the coordinates of point C.
|
[
{
"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 in the Cartesian coordinate system, there is a point A(0,1), and vectors $\\overrightarrow{AB}=(-4,-3)$, $\\overrightarrow{BC}=(-7,-4)$. Calculate the coordinates of point C. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
(-11,-6)
|
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