messages
listlengths 3
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[
{
"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 complex number $(a^2+2a-3) + (a^2+a-6)i$ represents a pure imaginary number, calculate the value of the real number $a$ such that the imaginary part is equal to zero.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 1 | 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": "On an island, there are knights who always tell the truth and liars who always lie. One day, 15 natives, among whom there were both knights and liars, formed a circle. Each of them said, \"Of the two people standing opposite me, one is a knight and the other is a liar.\" How many knights are there among them?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 10 | 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 function $y=f(x)$ is a continuous function on $[0,1]$, and $f(0) \\cdot f(1) < 0$, using the bisection method to find the zero of the function, the required accuracy of the approximate value reaches $0.1$, calculate the maximum number of times the interval needs to be halved.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 4 | 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 maximum number of points of intersection that can occur when $2$ different circles and $3$ different straight lines are drawn on the same piece of paper?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 17 | 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": "By adding the first two equations and subtracting the third, we obtain:\n\n\\[\n\\cos x + \\cos y + \\cos z + \\cos (x + y + z) = 0 \\Leftrightarrow\n\\]\n\n\\[\n\\Leftrightarrow 2 \\cos \\left(\\frac{x+y}{2}\\right) \\cdot \\cos \\left(\\frac{x-y}{2}\\right) + 2 \\cos \\left(\\frac{z+x+y+z}{2}\\right) \\cdot \\cos \\left(\\frac{y+x}{2}\\right) = 0 \\Leftrightarrow\n\\]\n\n\\[\n\\Leftrightarrow \\cos \\left(\\frac{x+y}{2}\\right) \\cdot \\left(\\cos \\left(\\frac{x-y}{2}\\right) + \\cos \\left(\\left(\\frac{x+y}{2}\\right) + z \\right) \\right) = 0 \\Leftrightarrow\n\\]\n\n\\[\n\\Leftrightarrow \\cos \\left(\\frac{x+y}{2}\\right) \\cdot \\cos \\left(\\frac{x+z}{2}\\right) \\cdot \\cos \\left(\\frac{y+z}{2}\\right) = 0 \\Leftrightarrow\n\\]\n\n\\[\n\\left[\\begin{array}{c}\n\\cos \\left(\\frac{x+y}{2}\\right) = 0 \\\\\n\\cos \\left(\\frac{x+z}{2}\\right) = 0 \\\\\n\\cos \\left(\\frac{y+z}{2}\\right) = 0\n\\end{array}\\right.\n\\]\n\n\\[\n\\left[\\begin{array}{c}\nx + y = \\pi + 2\\pi n \\\\\nx + z = \\pi + 2\\pi m \\\\\ny + z = \\pi + 2\\pi k, \\quad n, m, k \\in \\mathbb{Z}\n\\end{array}\\right.\n\\]\n\nSubstituting \\(x + y = \\pi + 2\\pi n\\) into the system, we obtain:\n\n\\[\n\\left\\{\\begin{array}{l}\n\\cos x - 2 \\cos z = 0 \\\\\n\\cos y - 3 \\cos z = 0 \\\\\n-5 \\cos z = 0\n\\end{array}\n\\Leftrightarrow \n\\left\\{\\begin{array}{l}\n\\cos x = 0 \\\\\n\\cos y = 0 \\\\\n\\cos z = 0\n\\end{array}\n\\Leftrightarrow\n\\left\\{\\begin{array}{l}\nx = \\frac{\\pi}{2} + \\pi n \\\\\ny = \\frac{\\pi}{2} + \\pi m \\\\\nz = \\frac{\\pi}{2} + \\pi k, \\quad n, m, k \\in \\mathbb{Z}\n\\end{array}\\right.\n\\right.\n\\right.\n\\]\n\nThe remaining cases are considered similarly.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | \left\{ \frac{\pi}{2} + \pi n ; \frac{\pi}{2} + \pi m ; \frac{\pi}{2} + \pi k \mid m, n, k \in \mathbb{Z} \right\} | 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 a tangent line $l$ to the curve $y=x^4$ is perpendicular to the line $x+4y-2009=0$, then the equation of the tangent line $l$ is.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 4x-y-3=0 | 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 number of ways to arrange the letters of the word PERMUTATIONS.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 239,500,800 | 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 square \\(ABCD\\) with side length 4 is folded along \\(BD\\) to form a dihedral angle of \\(60^{\\circ}\\). What is the distance between the midpoint of \\(BC\\) and point \\(A\\)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 2\sqrt{2} | 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 \\\\(l_{1}\\\\): \\\\(y=k_{1}x+b_{1}\\\\) and \\\\(l_{2}\\\\): \\\\(y=k_{2}x+b_{2}\\\\), the proposition \\\\(p\\\\): (\"If \\\\(l_{1} \\perp l_{2}\\\\), then \\\\(k_{1}k_{2}=-1\"\\\\)'s contrapositive is \\_\\_\\_\\_\\_\\_, and the original proposition \\\\(p\\\\) is a \\_\\_\\_\\_\\_\\_ proposition (fill in \"true\" or \"false\").\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | \text{If } k_{1}k_{2} \neq -1, \text{ then } l_{1} \text{ and } l_{2} \text{ are not perpendicular; true.} | 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 the complement of an angle is $30^{\\circ}$, then the degree of the supplement of this angle is ______.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 120^{\circ} | 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 summer, Ponchik eats honey cakes four times a day: instead of morning exercise, instead of a daytime walk, instead of an evening run, and instead of a nighttime swim. The quantities of cakes eaten instead of exercise and instead of a walk are in the ratio $3:2$; instead of a walk and instead of a run - in the ratio $5:3$; and instead of a run and instead of a swim - in the ratio $6:5$. How many more or fewer cakes did Ponchik eat instead of exercise compared to the swim on the day when a total of 216 cakes were eaten?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 60 | 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 group photo, 4 boys and 3 girls are to stand in a row such that no two boys or two girls stand next to each other. How many different arrangements are possible?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 144 | 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 sum of all integer solutions to $4 < (x-3)^2 < 36$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 18 | 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": "William is thinking of an integer between 1 and 50, inclusive. Victor can choose a positive integer $m$ and ask William: \"does $m$ divide your number?\", to which William must answer truthfully. Victor continues asking these questions until he determines William's number. What is the minimum number of questions that Victor needs to guarantee this?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 15 | 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 natural number that is equal to the sum of all preceding natural numbers. Does more than one such number exist, or is there only one?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 3 | 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 is a set of 6860 switches, each with five positions named $A, B, C, D, E$. The positions change cyclically from $A$ to $B$, $B$ to $C$, $C$ to $D$, $D$ to $E$, and $E$ back to $A$. Each switch is initially in position $A$. The switches are labeled with numbers of the form $(2^{x})(3^{y})(5^{z})(7^{w})$ where $x, y, z, w$ take on values $0, 1, \\ldots, 5$. For step $i$ of a 6860-step process, the $i$-th switch is advanced one step, along with all other switches whose labels divide the label on the $i$-th switch. After all steps are completed, how many switches will be in position $A$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 6455 | 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 the probability that a baby born in a certain hospital will speak in the next day is 1/3, what is the probability that at least 3 babies out of a cluster of 6 babies will speak tomorrow?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | \frac{233}{729} | 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 \\( \\frac{x}{y} = \\frac{12}{5} \\). If \\( y = 25 \\), what is the value of \\( x \\)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 60 | 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 $[x]$ denote the greatest integer less than or equal to $x$. Given that the natural number $n$ satisfies \\(\\left[\\frac{1}{15}\\right]+\\left[\\frac{2}{15}\\right]+\\left[\\frac{3}{15}\\right]+\\cdots+\\left[\\frac{\\mathrm{n}-1}{15}\\right]+\\left[\\frac{\\mathrm{n}}{15}\\right]> 2011\\), what is the smallest value of $n$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 253 | 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 $p$: $x^{2}+x-6=0$ is a necessary but not sufficient condition for $q$: $ax+1=0$, find the value of the real number $a$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | a=-\frac{1}{2} \text{ or } a=-\frac{1}{3} | 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": "Express the number \"$20.8$ billion in scientific notation.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 2.08 \times 10^9 | 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 $\\{a_n\\}$ is a geometric sequence with the sum of the first $n$ terms denoted as $S_n (n\\in\\mathbb{N^*})$, and $\\frac{1}{a_1}-\\frac{1}{a_2}=\\frac{2}{a_3}$, $(S_6=63)$.\n\n(1) Find the general term formula for $\\{a_n\\}$;\n(2) If for any $n\\in\\mathbb{N^*}$, $b_n$ is the arithmetic mean of $\\log_2{a_n}$ and $\\log_2{a_{n+1}}$, find the sum of the first $2n$ terms of the sequence $\\{(−1)^n b_n^2\\}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 2n^2 | 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 the function $f\\left(x\\right)=2^{x}+2^{-x}$, then the maximum value of $f\\left(x\\right)$ on the interval $[-\\frac{1}{2},\\frac{1}{2}]$ is ____.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 2 | 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 100 terms of the sequence $\\{a_n\\}$, where the general term formula is $a_n = (-1)^{n-1} \\cdot (4n-3)$, equals what value?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | -200 | 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 \\( x + \\frac{1}{x} = 10 \\), then what is the value of \\( x^2 + \\frac{1}{x^2} \\)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 98 | 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 $x$ be a real number, $x > 1.$ Compute the infinite sum:\n\\[\n\\sum_{n = 0}^\\infty \\frac{1}{x^{3^n} - x^{-3^n}}.\n\\]\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | \frac{1}{x - 1} | 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 smallest possible area, in square units, of a right triangle with sides measuring $6$ units and $8$ units?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 15.87 | 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 an opaque bag, there are $2$ green balls, $3$ black balls, and $6$ red balls, all identical except for color. The probability of randomly drawing a green ball from the bag is ______.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | \frac{2}{11} | 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": "Bonnie constructs the frame of a cube using 12 pieces of wire, each 8 inches long. Roark creates several separate unit cube frames using 1-inch-long pieces of wire. The total volume of Roark's cubes matches the volume of Bonnie's cube. Determine the ratio of the total length of Bonnie's wire to Roark's wire. Express your answer as a common fraction.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | \frac{1}{64} | 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 $x$, $y$, and $z$ are positive with $xy=30$, $xz = 60$, and $yz=90$, what is the value of $x+y+z$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | x+y+z = 11\sqrt{5} | 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 points $A(2,3)$, $B(5,4)$, $C(7,10)$, if point $P$ satisfies $\\overrightarrow{AP}=\\overrightarrow{AB}+λ\\overrightarrow{AC}$ ($λ∈R$), and point $P$ is in the third quadrant, then the range of $λ$ is _______.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | (-∞,-1) | 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": "Samantha believes that the amount of time she studies for a test and her score on that test are directly related. On her first exam, she studied for three hours and scored 60 on the exam. How many hours does Samantha believe she must study for her second exam so that the average of her two exam scores is 75?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 4.5 | 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 numbers $x, y, z, w$ are equal to 1, 5, 2, 4, in some order. Find the largest possible value of\n\\[xy - yz + zw - wx.\\]\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 9 | 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 an arithmetic sequence $\\{a\\_n\\}$ with the sum of the first $n$ terms denoted as $S\\_n$, if $S\\_8 = 2S\\_4$, find the value of $\\frac{a\\_3}{a\\_1}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | \frac{a\_3}{a\_1} = 1 | 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 residue of $7^{1234}$, modulo 13?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 12 | 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 the area of the shaded region is $\\frac{32}{\\pi}$, and the radius of the smaller circle is three times smaller than the radius of the larger circle. What is the circumference of the smaller circle?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 4 | 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 \\\\(i\\\\) is the imaginary unit, simplify the expression \\\\(\\dfrac{1+i}{3-i}\\\\.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | \frac{1+2i}{5} | MATH |
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Compute\n\\[\n\\frac{\\lfloor \\sqrt{1} \\rfloor \\cdot \\lfloor \\sqrt{3} \\rfloor \\cdot \\lfloor \\sqrt{5} \\rfloor \\dotsm \\lfloor \\sqrt{35} \\rfloor}{\\lfloor \\sqrt{2} \\rfloor \\cdot \\lfloor \\sqrt{4} \\rfloor \\cdot \\lfloor \\sqrt{6} \\rfloor \\dotsm \\lfloor \\sqrt{36} \\rfloor}.\n\\]\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | \frac{1}{6} | 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": "Eight times $A$'s money added to $B$'s money is more than $160.00$. Four times $A$'s money plus $B$'s money is $120.00$.\nIf $a$ represents $A$'s money in dollars and $b$ represents $B$'s money in dollars, then:\n**A)** $a > 10$, $b > 80$ \n**B)** $a > 10$, $b < 80$ \n**C)** $a > 10$, $b = 80$ \n**D)** $a > 10$, and as $a$ increases, $b$ decreases \n**E)** $2a = 3b$\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | \textbf{(D)}\ a > 10, and as a increases, b decreases | 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 sphere with center $O$ has a radius of $5$. An isosceles triangle with sides of length $13, 13,$ and $10$ is situated in space so that each of its sides is tangent to the sphere. Calculate the distance from $O$ to the plane determined by the triangle.\n**A)** $\\frac{5\\sqrt{5}}{3}$\n**B)** $\\frac{10}{3}$\n**C)** $\\frac{5\\sqrt{3}}{3}$\n**D)** $5\\sqrt{2}$\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | \textbf{(A)} \frac{5\sqrt{5}}{3} | 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 \\( t(x) = \\sqrt{4x + 1} \\) and \\( f(x) = 7 - t(x) \\). What is \\( t(f(5)) \\)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | \sqrt{29 - 4\sqrt{21}} | 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 two distinct numbers are selected at random from the first ten prime numbers, what is the probability that their sum is an even number? Express your answer as a common fraction.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | \frac{4}{5} | 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 stock investor purchased a certain stock, which then experienced 3 consecutive down limits followed by 3 consecutive up limits. What is the net percentage change of the stock price?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 0.970299 | 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 sum of all positive integers $n$ such that, given an unlimited supply of stamps of denominations $6, n,$ and $n+1$ cents, $115$ cents is the greatest postage that cannot be formed.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 24 | 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 $A = 3 - 2i$, $M = -3 + 2i$, $S = 2i$, $P = -1$, and $R = A \\cdot M$, compute the value of $A - M + S - P + R$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 2 + 10i | 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 people, player A and player B, are playing a game of chess. The probability of them drawing the game is $\\frac{1}{2}$, and the probability of player B winning is $\\frac{1}{3}$. What is the probability that player A does not lose?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | \frac{2}{3} | 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 chess tournament, \\( n \\) women and \\( 2n \\) men participated. Everyone played exactly one game with each other. There were no draws, and the number of games won by the women is in the ratio of \\( 7: 5 \\) to the number of games won by the men. What is \\( n \\)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 3 | 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 the one-variable quadratic equation $x^{2}+2x+m+1=0$ has two distinct real roots with respect to $x$, determine the value of $m$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | -1 | 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": "$n$ is a natural number. If $9n^{2}+5n-26$ is the product of two consecutive natural numbers, then the value of $n$ is ______.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 2 | 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": "On a calm evening, Linus observes a flash of lightning and hears the thunder 15 seconds later. Considering the speed of sound is 1100 feet per second and 1 kilometer is approximately 3280 feet, estimate the distance Linus was from the lightning to the nearest quarter-kilometer.\nA) 4.75 kilometers\nB) 5 kilometers\nC) 5.25 kilometers\nD) 5.5 kilometers\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | B) 5 kilometers | 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 a sequence $\\{a_n\\}$ whose sum of the first $n$ terms $S_n = 2^n - 1$, calculate the value of $a_3$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 4 | 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 plane vectors $\\overrightarrow{a}$, $\\overrightarrow{b}$, $\\overrightarrow{c}$ satisfy $\\langle \\overrightarrow{a}, \\overrightarrow{b} \\rangle = 60^\\circ$, and $\\{|\\overrightarrow{a}|, |\\overrightarrow{b}|, |\\overrightarrow{c}|\\} = \\{1, 2, 3\\}$, calculate the maximum value of $|\\overrightarrow{a}+ \\overrightarrow{b}+ \\overrightarrow{c}|$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | \sqrt{7}+3 | 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 digits 1, 3, 5, and 9 are arranged randomly to form a four-digit number. What is the probability that the number is odd? Express your answer as a common fraction.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 1 | 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": "Say a positive integer $n$ is *radioactive* if one of its prime factors is strictly greater than $\\sqrt{n}$ . For example, $2012 = 2^2 \\cdot 503$ , $2013 = 3 \\cdot 11 \\cdot 61$ and $2014 = 2 \\cdot 19 \\cdot 53$ are all radioactive, but $2015 = 5 \\cdot 13 \\cdot 31$ is not. How many radioactive numbers have all prime factors less than $30$ ?\n\n*Proposed by Evan Chen*\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 119 | 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 length of the real axis of a hyperbola is $m$, and the distance from a point $P$ on this hyperbola to the right focus is also $m$. The distance from point $P$ to the left focus of this hyperbola is.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 2m | 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": "Three balls, a red, a green, and a blue ball, are tossed independently into bins numbered with the positive integers such that the probability a ball lands in bin $k$ is $3^{-k}$ for $k=1,2,3,...$. Calculate the probability that all three balls land in different bins.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | \frac{25}{26} | 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 set $A=\\{x||x-1| \\lt 1\\}$, set $B=\\left\\{{x|{y=\\sqrt{2-x}}}\\right\\}$, find the intersection of sets A and B.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | (0,2) | 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 full score for this question is 12 points) Let the function $f(x) = ax^3 + bx + c$ ($a \\neq 0$) be an odd function. The tangent line at the point $(1, f(1))$ is perpendicular to the line $x - 6y - 7 = 0$, and the minimum value of the derivative $f'(x)$ is $-12$.\n(1) Find the values of $a$, $b$, and $c$;\n(2) Find the intervals of monotonic increase for the function $f(x)$, and find the maximum and minimum values of the function $f(x)$ on the interval $[-1,3]$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 18 \text{ and } -8 | 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 $(\\frac{1}{2})^{x} \\leqslant 4$ and $\\log_{\\sqrt{3}}x \\leqslant 2$, find the maximum and minimum values of the function $f(x)=9^{x}-3^{x+1}-1$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | -\frac{13}{4} | 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 $ S$ be a set of all points of a plane whose coordinates are integers. Find the smallest positive integer $ k$ for which there exists a 60-element subset of set $ S$ with the following condition satisfied for any two elements $ A,B$ of the subset there exists a point $ C$ contained in $ S$ such that the area of triangle $ ABC$ is equal to k .\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 210 | 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 the function $f(x)$ is differentiable on $\\mathbb{R}$ and $f(x)=x^{2}+2f'(1)x+3$, determine the relationship between $f(0)$ and $f(4)$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | f(0)=f(4) | 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\n\\[ f(x) = \\sqrt{x(100 - x)} + \\sqrt{x(4 - x)} \\]\nfor \\( 0 \\le x \\le 4 \\). Find the maximum value \\( M \\) of \\( f(x) \\), and determine the value \\( x = x_0 \\) at which this maximum occurs. Enter the ordered pair \\( (x_0, M) \\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | \left(4, 8\sqrt{6}\right) | 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 hundreds digit of $(25! - 20!)$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 0 | 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 all positive integer pairs \\((a, b)\\) that make \\(a b^{2}+b+7\\) divide \\(a^{2} b+a+b\\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | (11, 1), (49, 1), (7k^2, 7k) \text{ where } k \in \mathbf{N}^* | 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 number is called *purple* if it can be expressed in the form $\\frac{1}{2^a 5^b}$ for positive integers $a > b$ . The sum of all purple numbers can be expressed as $\\frac{a}{b}$ for relatively prime positive integers $a, b$ . Compute $100a + b$ .\n\n*Proposed by Eugene Chen*\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 109 | 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 the domain of the function $f(x)$ is $R$, $f(x) > f'(x) + 1$, and $f(0) = 3$, determine the solution set of the inequality $f(x) > 2e^{x} + 1$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | (-\infty, 0) | 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": "Jo and Blair take turns counting, with each number being two more than the last number said by the other person, starting from 3. Find the 40th number said.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 81 | 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 is a wooden stick with a length of $11$ meters. What is the probability that if it is broken into two pieces, both pieces are no less than $3$ meters in length?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | \dfrac{5}{11} | 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 school level magic and spellcasting competition consists of 5 spells. Out of 100 young wizards who participated in the competition:\n\n- 95 performed the 1st spell correctly\n- 75 performed the 2nd spell correctly\n- 97 performed the 3rd spell correctly\n- 95 performed the 4th spell correctly\n- 96 performed the 5th spell correctly.\n\nWhat is the minimum number of students who could have performed exactly 4 out of the 5 spells correctly under the given conditions?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 8 | 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": "Billy and Bobbi each selected a positive integer less than 500. Billy's number is a multiple of 20, and Bobbi's number is a multiple of 30. What is the probability that they selected the same number? Express your answer as a common fraction.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | \frac{1}{50} | 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": "Each natural number from 1 to 50000 is replaced by the sum of its digits. This operation is repeated with the resulting numbers until all numbers become single-digit. How many times will each of the integers from 0 to 8 appear among these single-digit numbers?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 5555 | 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 number of proper subsets of the set {1, 2, 3} is .\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 7 | 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 rectangle $ABCD$, $AB=BC=15$, $CD=DA=20$, and $\\angle D=90^\\circ$. Points $X$ and $Y$ are the midpoints of $\\overline{BC}$ and $\\overline{DA}$ respectively. Compute $XY^2$ (the square of the length of $XY$).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 400 | 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 an arithmetic sequence $\\{a_n\\}$ and a geometric sequence $\\{b_n\\}$ satisfying: $a_1 = b_1 = 1$, $a_2 = b_2$, and $2a_3 - b_3 = 1$.\n(Ⅰ) Find the general formulas for the sequences $\\{a_n\\}$ and $\\{b_n\\}$;\n(Ⅱ) Denoting $c_n = a_n b_n$, calculate the sum $S_n$ of the first $n$ terms of the sequence $\\{c_n\\}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | \frac{(n-1)3^n + 1}{2} | 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 $\\overrightarrow{AP}= \\frac{1}{3} \\overrightarrow{PB}$ and $\\overrightarrow{AB}=λ \\overrightarrow{BP}$, determine the value of λ.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | -\frac{4}{3} | 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 John paid $15 for four muffins under a promotional sale that offers \"Buy three muffins at the regular price and get the fourth muffin for free\", determine the regular price of one muffin.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 5 | 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 triangle with sides of length 10, 11, and 13, point $H$ is the orthocenter. If $AD$ is the altitude to the side of length 11, find the ratio $HD:HA$, where $D$ and $A$ are as previously defined with $A$ opposite the side of length 11.\nA) 0:1\nB) 1:2\nC) 2:3\nD) 1:3\nE) 1:4\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | The final answer is **A) 0:1**. | 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": "Dots are spaced one unit apart, both horizontally and vertically. What is the number of square units enclosed by the following polygon formed by these points?\n[asy]\nsize(1inch,1inch);\npair a=(0,0), b=(20,0), c=(30,0), d=(40, 0);\npair e=(0,10), f=(20,10), g=(30,10), h=(40,10);\npair i=(0,20), j=(20,20), k=(30,20), l=(40,20);\npair m=(0,30), n=(20,30), o=(30,30), p=(40,30);\ndot(a);\ndot(b);\ndot(c);\ndot(d);\ndot(e);\ndot(f);\ndot(g);\ndot(h);\ndot(i);\ndot(j);\ndot(k);\ndot(l);\ndot(m);\ndot(n);\ndot(o);\ndot(p);\ndraw(a--b--g--c--d--h--l--o--n--m--i--e--a);\n[/asy]\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 7.5 | 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 $2a=3b$, calculate the value of $\\frac{a}{b}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | \frac{3}{2} | 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 $x=1$ is a local minimum point of the function $f(x)=(x-2)e^{x}- \\frac{k}{2}x^{2}+kx$ ($k > 0$), the range of the real number $k$ is ______.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | \{k|0 < k < e\} | 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 woman buys an apartment for $20,000 and rents it out. She sets aside 10% of each month's rent for maintenance and pays $460 a year in taxes. Determine the monthly rent she needs to charge to realize a 6% return on her investment annually.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 153.70 | 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 the function $f(x)=x^{3}+(1-a)x^{2}-a(a+2)x+b$ $(a,b\\in\\mathbb{R})$.\n$(1)$ If the graph of the function $f(x)$ passes through the origin and the slope of the tangent line at the origin is $-3$, find the values of $a$ and $b$;\n$(2)$ If the curve $y=f(x)$ has two tangents perpendicular to the $y$-axis, find the range of values for $a$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | (-\infty,- \frac {1}{2})\cup(- \frac {1}{2},+\infty) | 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 sum of the positive integer divisors of 37?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 38 | 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": "Proposition $p$: The inequality $x^2 + 2ax + 4 > 0$ holds true for all real numbers $x$; Proposition $q$: The function $f(x) = (3 - 2a)^x$ is an increasing function. If either $p$ or $q$ is true, and $p$ and $q$ are not both true, find the range of values for the real number $a$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | a \leq -2 \text{ or } 1 \leq a < 2 | 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 $x$: $$ \\frac{1}{3} - \\frac{1}{4} = \\frac{1}{x}.$$\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 12 | 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 the central angle of a sector is ${{120}^{0}}$, and the arc length is $2cm$, then the area of this sector equals \\_\\_\\_\\_\\_\\_\\_$cm^2$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | \dfrac{3}{\pi} | 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 the function be defined as $y = \\frac{15x^4 + 6x^3 + 7x^2 + 4x + 5}{5x^5 + 3x^3 + 9x^2 + 2x + 4}$. Determine if there is a horizontal asymptote and if so, find its $y$-value. Also, calculate the x-intercept(s) of the function.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | y = 0 | 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 the equations $3x + 2y = 8$ and $2x + 3y = 11$, find the value of $10x^2 + 13xy + 10y^2$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | \frac{2041}{25} | 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": "Cedric has deposited $\\$15,\\!000$ into an account that pays $6\\%$ interest compounded annually.\n\nDaniel has deposited $\\$15,\\!000$ into an account that pays $8\\%$ simple annual interest.\n\nAfter $20$ years, how much more money does Cedric have in his account compared to Daniel, to the nearest dollar?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | \$9107 | 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 graph of the function $g(x)$ is shown below. How many values of $x$ satisfy $g(g(x)) = 2$? [asy]\nimport graph; size(7.4cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.4,xmax=5.66,ymin=-1.05,ymax=6.16;\n\nfor(int i = -4; i <= 5; ++i) {\n draw((i,-1)--(i,6), dashed+mediumgrey);\n}\n\nfor(int i = 1; i <= 6; ++i) {\n draw((-4,i)--(5,i), dashed+mediumgrey);\n}\n\nLabel laxis; laxis.p=fontsize(10);\n\nxaxis(\"$x$\",-4.36,5.56,defaultpen+black,Ticks(laxis,Step=1.0,Size=2,OmitTick(0)),Arrows(6),above=true); yaxis(\"$y$\",-0.92,6.12,defaultpen+black,Ticks(laxis,Step=1.0,Size=2,OmitTick(0)),Arrows(6),above=true); draw((xmin,(-(0)-(-2)*xmin)/-2)--(-1,(-(0)-(-2)*-1)/-2),linewidth(1.2)); draw((-1,1)--(3,5),linewidth(1.2)); draw((3,(-(-16)-(2)*3)/2)--(xmax,(-(-16)-(2)*xmax)/2),linewidth(1.2)); // draw((min,(-(-9)-(0)*xmin)/3)--(xmax,(-(-9)-(0)*xmax)/3),linetype(\"6pt 6pt\"));\n\nlabel(\"$g(x)$\",(-3.52,4.6),SE*lsf);\n\n//dot((-1,1),ds); dot((3,5),ds); dot((-3,3),ds); dot((1,3),ds); dot((5,3),ds);\n\ndot((-4.32,4.32),ds); dot((5.56,2.44),ds);\nclip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);\n\n[/asy]\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 2 | 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 a sequence $\\{a_{n}\\}$ where the first term is $a_{1}=1$, and ${a_{n+1}}=\\left\\{\\begin{array}{l}{a_n}+2$, when $n$ is odd, $3{a_n}$, when $n$ is even$\\end{array}\\right.$, then the sum of the first $2n$ terms of the sequence $\\{a_{n}\\}$ is ____.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 4\times 3^{n} - 4n - 4 | 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 $a$, $b$, and $c$ are the sides of a right-angled triangle, where $c$ is the hypotenuse, if the inequality $\\frac{1}{a^{2}} + \\frac{4}{b^{2}} + \\frac{t}{c^{2}} \\geqslant 0$ always holds true, then the range of values for the real number $t$ is _______.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | [-9, +\infty) | 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 distance from point \\( M_{0} \\) to the plane passing through the three points \\( M_{1}, M_{2}, M_{3} \\).\n\n\\[ M_{1}(1, 2, 0) \\]\n\n\\[ M_{2}(3, 0, -3) \\]\n\n\\[ M_{3}(5, 2, 6) \\]\n\n\\[ M_{0}(-13, -8, 16) \\]\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | \frac{134}{7} | 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 function $g(x)$ as:\n\\[\ng(x) =\n\\begin{cases}\nx^2 - 1 &\\quad \\text{if } x \\ge -2, \\\\\nx + 5 &\\quad \\text{otherwise},\n\\end{cases}\n\\]\nDetermine the number of values of $x$ for which $g(g(x)) = 9$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | 0 | 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": "Megan will roll two six-sided dice. What is the probability that she will roll an even number on the first die and an odd number on the second die? Express your answer as a common fraction.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | \frac{1}{4} | 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 the sum of the first $n$ terms of the sequence $\\{a_{n}\\}$ is $S_{n}$, $a_{1}=2$, and ${S_n}{S_{n-1}}+{a_n}=0$ for $n≥2$, $n∈{N^*}$, find $S_{n}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | \frac{2}{2n-1} | 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 \\( |\\overrightarrow{AB}| = 10 \\). If a point \\( P \\) on the plane satisfies \\( |\\overrightarrow{AP} - t \\overrightarrow{AB}| \\geq 3 \\) for any \\( t \\in \\mathbb{R} \\), then the minimum value of \\( \\overrightarrow{PA} \\cdot \\overrightarrow{PB} \\) is \\(\\qquad\\), and at this point, \\( |\\overrightarrow{PA} + \\overrightarrow{PB}| = \\) \\(\\qquad\\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | -16, 6 | 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 $f(x)$ satisfies for all $x \\in \\mathbb{R}$, $f(-x) + f(x) = 0$, and when $x \\geq 0$, $f(x) = e^x + m$ (where $m$ is a constant), calculate the value of $f(-\\ln 5)$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | -4 | 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 $n$ be a positive integer. Find the maximal real number $k$ , such that the following holds:\n\nFor any $n$ real numbers $x_1,x_2,...,x_n$ , we have $\\sqrt{x_1^2+x_2^2+\\dots+x_n^2}\\geq k\\cdot\\min(|x_1-x_2|,|x_2-x_3|,...,|x_{n-1}-x_n|,|x_n-x_1|)$ \n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | \frac{\sqrt{n}}{2} \text{ if } n \text{ is even and } \frac{\sqrt{n+5}}{2} \text{ if } n \text{ is odd} | 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 vectors $\\overrightarrow{a}=(\\sin x, \\cos x)$, $\\overrightarrow{b}=(\\sin x, \\sin x)$, $\\overrightarrow{c}=(-1, 0)$. \n(1) If $x= \\frac{\\pi}{3}$, determine the angle $\\theta$ between vectors $\\overrightarrow{a}$ and $\\overrightarrow{c}$. \n(2) If $x \\in \\left[-\\frac{3\\pi}{8}, \\frac{\\pi}{4}\\right]$, and the function $f(x)=\\lambda \\overrightarrow{a} \\cdot \\overrightarrow{b}$ has a maximum value of $\\frac{1}{2}$, find the real number $\\lambda$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
},
{
"content": "I am a dummy message",
"role": "user"
}
] | \frac{1}{2} | MATH |