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eurus2_ground_truth_with_system

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AI2 Adapt Dev 46

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 $n$ modulo 9, where $$n = 2+33+444+5555+66666+777777+8888888$$ and $0 \\le n < 9$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 management staff caught 30 fish from a pond, marked them, and then released them back into the pond. 10 days later, they caught 50 fish from the pond again, among which 2 were marked. Based on the data provided, estimate the total number of fish in the pond.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 arithmetic sequence $\\{a\\_n\\}$ has a first term of $1$ and the sum of its first $n$ terms is $S\\_n$, with $S\\_5 = 20$, find the common difference.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 $-3 - 5\\sqrt{3}$ is a root of the equation \\[x^3 + ax^2 + bx + 90 = 0\\] and that $a$ and $b$ are rational numbers, compute $a.$\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 $f(x) = -\\frac{1}{3}x^3 + \\frac{1}{2}x^2 + 2ax$.\n(1) If $f(x)$ has a strictly increasing interval on $(\\frac{2}{3}, +\\infty)$, find the range of values for $a$.\n(2) When $a=1$, find the maximum and minimum values of $f(x)$ on the interval $[1,4]$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 least positive integer $n$ for which $\\frac{n-17}{7n+4}$ is a non-zero reducible fraction.\nA) 20\nB) 58\nC) 75\nD) 99\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Suppose there are 3 counterfeit coins of equal weight mixed with 12 genuine coins. All counterfeit coins weigh differently from the genuine coins. A pair of coins is selected at random without replacement from the 15 coins, followed by selecting a second pair from the remaining 13 coins. The combined weight of the first pair equals the combined weight of the second pair. What is the probability that all four coins selected are genuine?\nA) $\\frac{7}{11}$\nB) $\\frac{9}{13}$\nC) $\\frac{11}{15}$\nD) $\\frac{15}{19}$\nE) $\\frac{15}{16}$\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 sequence $\\{a_n\\}$ with the sum of the first $n$ terms $S_n=1-5+9-13+17-21+......+{(-1)}^{n-1}(4n-3)$, calculate the value of $S_{15}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 $\\int_{0}^{n}{|x-5|\\mathrm{d}x}=25$, then the coefficient of $x^{2}$ in the binomial expansion of $(2x-1)^{n}$ is __________.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 first four terms in an arithmetic sequence are given as $x+2y$, $x-y$, $2xy$, and $\\frac{x}{2y}$. Determine the fifth term and 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" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 coordinate system, trapezoid $WXYZ$ has vertices $W(2, 3)$, $X(6, 3)$, $Y(7, -1)$, and $Z(3, -1)$. Determine the area of trapezoid $WXYZ$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 $\\left(\\frac{10}{11}\\right)^3 \\cdot \\left(\\frac{1}{3}\\right)^3$ is calculated and then added to $\\left(\\frac{1}{2}\\right)^3$, what is the resulting value?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 integers, $x$, satisfy the inequality $|7x - 5| \\le 9$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 $f(x)=2\\ln 3x+8x$, find the value of $\\lim_{\\triangle x\\rightarrow 0} \\frac {f(1-2\\triangle x)-f(1)}{\\triangle x}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 volume of the tetrahedron with vertices $A_{1}(2,3,1)$, $A_{2}(4,1,-2), A_{3}(6,3,7)$, and $A_{4}(-5,-4,8)$ and its height dropped from vertex $A_{4}$ to the face $A_{1} A_{2} A_{3}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 O(0,0) and A(1,1), the slope angle of line OA is $\\boxed{\\text{answer}}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 certain supermarket currently has 20 baskets of tomatoes, with a standard weight of 18 kilograms per basket. The excess or shortfall in kilograms is represented by positive or negative numbers, as shown below:\n\n| Difference from Standard Weight (kg) | -2.5 | -2 | -0.5 | 0 | 1 | 1.5 |\n|--------------------------------------|------|----|------|---|---|-----|\n| Number of Baskets | 2 | 4 | 2 | 1 | 3 | 8 |\n\n$(1)$ Compared to the standard weight, how many kilograms are the 20 baskets of tomatoes in total over or underweight?<br/>$(2)$ The supermarket participates in the \"Benefiting the People and Nourishing the Heart\" project, and tomatoes are sold at a price of 7 yuan per kilogram. How much yuan can these 20 baskets of tomatoes be sold for?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 solution set of the inequality $|2-x| \\geq 1$ can be expressed as an interval.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 two islands, $A$ and $B$, in the sea that are $10\\,n\\text{ mile}$ apart. The angle formed by looking at island $C$ from island $A$ and island $B$ is $60^{\\circ}$, and the angle formed by looking at island $C$ from island $B$ and island $A$ is $75^{\\circ}$. What is the distance between island $B$ and island $C$ in $n\\text{ mile}$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 P (2, 0), and circle C: $x^2+y^2-6x+4y+4=0$. When line L passes through point P and the distance from the center of circle C is 1, find the equation of line L.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a heart ($\\heartsuit$) and the second card is a club ($\\clubsuit$)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 $f(x)=2\\sin (x- \\frac {π}{6})\\sin (x+ \\frac {π}{3})$, $x\\in R$.\n(I) Find the smallest positive period of the function $f(x)$;\n(II) In $\\triangle ABC$, if $A= \\frac {π}{4}$, and acute angle $C$ satisfies $f( \\frac {C}{2}+ \\frac {π}{6})= \\frac {1}{2}$, find the value of $\\frac {BC}{AB}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 $A=\\{x\\mid-1 < x < 2\\}$ and $B=\\{x\\mid2a-1 < x < 2a+3\\}$.\n1. Find the range of values for $a$ if $A\\subseteq B$.\n2. Find the range of values for $a$ if $A\\cap B=\\varnothing$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 $0 < α < \\frac{\\pi }{2} < β < π$, $\\tan \\frac{α}{2}= \\frac{1}{2}$, and $\\cos (β-α)= \\frac{ \\sqrt{2}}{10}$, find:\n\n(1) The value of $\\frac{\\cos 2\\alpha }{\\sqrt{2}\\cos (\\dfrac{\\pi }{4}+\\alpha )\\sin \\alpha }$\n\n(2) The value of $β$\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 $f(x)=2\\cos(2x+\\varphi)$, $(|\\varphi| < \\frac{\\pi}{2})$, find the minimum value of the function $f(x)$ in the interval $x\\in[0, \\frac{\\pi}{2}]$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$, respectively, and $(2a-c)\\cos B=b\\cos C$.\n(I) Find the measure of angle $B$;\n(II) If $\\cos A=\\frac{\\sqrt{2}}{2}$ and $a=2$, find the area of $\\triangle ABC$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 $x^2 + y^2 = m^2$ and $x + y = m$, determine the value of $m$ such that the graph of $x^2 + y^2 = m^2$ is tangent to the graph of $x + y = m$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 integer whose cube is equal to the sum of the cubes of the three preceding consecutive numbers.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 interval within which the solution to the equation $f(x) = 0$ lies, given the function $f(x) = 2^x + x - 5$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 $y=\\tan(2x- \\frac{\\pi}{4})$, determine the interval for which the function is monotonic increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 product inspection, the method of sampling inspection is often used. Now, 4 products are randomly selected from 100 products (among which there are 3 defective products) for inspection. The number of ways to exactly select 2 defective products is ____. (Answer with a number)\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 $\\overrightarrow{a}=(\\sin 2x,\\cos 2x)$ and $\\overrightarrow{b}=\\left( \\dfrac{\\sqrt{3}}{2}, -\\dfrac{1}{2} \\right)$, let $f(x)= \\overrightarrow{a} \\cdot \\overrightarrow{b}$.\n$(1)$ Find the period of the function $f(x)$;\n$(2)$ In $\\triangle ABC$, if $f(A)= \\dfrac{1}{2}$, $AB=2\\sqrt{3}$, $BC=2$, find the area $S$ of $\\triangle ABC$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 area of one lateral face of a right pyramid with an equilateral triangular base is 120 square meters. The slant height of the pyramid is 40 meters. Determine the length of the side of its base, in meters.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 survey of regional divisions, the distribution of village populations is represented in a pie chart. What percent of the villages have fewer than 50,000 inhabitants?\n\n- 45% of villages have populations from 20,000 to 49,999 residents.\n- 30% of villages have fewer than 20,000 residents.\n- 25% of villages have 50,000 or more residents.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 prime numbers have squares between 5000 and 9000?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 diagonals of a trapezoid are mutually perpendicular, and the midline is equal to 5. Find the segment connecting the midpoints of the bases.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 $\\{1, 2\\} \\subseteq X \\subseteq \\{1, 2, 3, 4, 5\\}$, calculate the total number of sets $X$ that satisfy this relation.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 an acute angle $\\alpha$ satisfies $\\tan 2\\alpha = -\\frac{4}{3}$, then $\\sin(\\pi - \\alpha) =$ ____.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 units digit of $(5619 + 2272)^{124}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 complex number \\( z = 1 + \\mathrm{i} + \\frac{m}{1 + \\mathrm{i}} \\) (where \\( m \\) is a real number) has both its real part and imaginary part greater than 0. Determine the range of values for \\( m \\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Given two complex numbers $z_1$ and $z_2$ whose corresponding points in the complex plane are symmetric with respect to the line $y = x$. If $z_1 = 1 + 2i$, calculate the value of $\\frac{z_1}{z_2}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 set of points that are inside or within one unit of a rectangular parallelepiped that measures $2$ by $3$ by $4$ units. Find the volume of this set, given that the volume can be expressed as $\\frac{m + n\\pi}{p},$ where $m, n,$ and $p$ are positive integers, and $n$ and $p$ are relatively prime. Determine $m + n + p.$\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 first container is $\\tfrac{3}{4}$ full of juice and the second container is initially empty, if the juice is poured from the first container into the second container, filling it to $\\tfrac{2}{3}$ of its total volume, calculate the ratio of the volume of the first container to the volume of the second container.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 problem involves finding the value of the expressions $\\lg 2 + \\lg 5$ and $4(-100)^4$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "When $a=$____, $3x^{a-1}-x=5$ is a quadratic equation in terms of $x$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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(0,0) \\), \\( B(1,0) \\), \\( C(0,2) \\), and \\( D(3,3) \\), point \\( P \\) is any point in the plane. Determine the minimum value of \\( |PA| + |PB| + |PC| + |PD| \\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 $f(x)= \\frac{ax}{x^{2}+b}(a > 0,b > 1)$, it is known that: $f(1)=1$, and $f(x)$ has a maximum value of $\\frac{3\\sqrt{2}}{4}$ on $R$.\n(I) Find the expression of $f(x)$;\n(II) When $x \\in [1,2]$, the inequality $f(x) \\leqslant \\frac{3m}{(x^{2}+2)|x-m|}$ always holds. Find the range of values for the real number $m$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 new map of a botanical garden uses a scale where one inch represents 500 feet. A line segment on this map measures 6.25 inches. How many yards does this line segment represent in the actual garden?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 imaginary part of the complex number $i(1-i)$ is to be found.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 $ABCD$ and $BCFG$ be two faces of a cube with $AB=10$. A beam of light emanates from vertex $A$ and reflects off face $BCFG$ at point $P$, which is 6 units from $\\overline{BG}$ and 4 units from $\\overline{BC}$. The beam continues to be reflected off the faces of the cube. Find the total length of the light path from the time it leaves point $A$ until it next reaches a vertex of the cube, in the form $m\\sqrt{n}$, where $m$ and $n$ are integers and $n$ is not divisible by the square of any prime.\n\n---\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 positive integer that is both a multiple of $9$ and a multiple of $6$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 $f(x)= \\sqrt{3} \\sin 2x+\\cos 2x-1(x\\in R)$;\n(1) Write out the smallest positive period and the monotonically increasing interval of the function $f(x)$;\n(2) In $\\triangle ABC$, angles $A$, $B$, $C$ correspond to sides $a$, $b$, $c$, respectively. If $f(B)=0$, $\\overrightarrow{BA}\\cdot \\overrightarrow{BC}= \\frac{3}{2}$, and $a+c=4$, find the value of $b$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 polar coordinates of the center of the circle $\\rho=2\\cos \\theta+2\\sin \\theta$ are $(r, \\theta)$, where $r$ and $\\theta$ are real numbers. Determine the values of $r$ and $\\theta$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 Ministry of Education of China, in order to develop education in poor areas, trains education major students for free in key normal universities across the country. Upon graduation, these students are assigned to teach in corresponding regions. Currently, there are 6 graduates who need to be evenly distributed to 3 schools. There are $\\boxed{\\text{answer}}$ different ways to assign them.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Arina wrote down all the numbers from 71 to 81 consecutively without spaces, obtaining the large number 717273...81. Sophia then continued appending the next numbers (i.e., first she appended 82, then 83, and so on). She stopped at the moment when the large number became divisible by 12. The last number she appended was $N$. What is the value of $N$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 the ellipse $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$, a point $P(3,y)$ on it has distances to the two foci that are $6.5$ and $3.5$, respectively. Find the equation of the ellipse.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": " $a,b,c,x,y,z$ are positive real numbers and $bz+cy=a$ , $az+cx=b$ , $ay+bx=c$ . Find the least value of following function $f(x,y,z)=\\frac{x^2}{1+x}+\\frac{y^2}{1+y}+\\frac{z^2}{1+z}$ \n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 integers \\( x \\) such that \\( x^{2}+x+13 \\) is a multiple of 121.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Thirty percent less than 90 is one-fourth more than which number?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 total of 32 students from Hope Elementary School participated in both Go and Chess competitions. Among them, 18 students participated in the Go competition, and 23 students participated in the Chess competition. The number of students who participated in both the Go and Chess competitions is ________.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Marty wants to paint a box. He can choose to use either blue, green, yellow, black, or white paint. Additionally, he can choose his painting method from a brush, roller, sponge, or spray. How many different combinations of color and painting method can Marty choose?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 $a=C_{20}^{0}+C_{20}^{1}\\cdot 2+C_{20}^{2}\\cdot {{2}^{2}}+\\cdots +C_{20}^{20}\\cdot {{2}^{20}}$ and $a \\equiv b \\pmod{10}$, find the value of $b$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "We define a number as a valley number if its middle digit is smaller than any other digit. For example, 495 is a valley number. How many 3-digit valley numbers are there?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 analytical expressions of the following functions:\n1. Given $f(x)=x^{2}+2x$, find $f(2x+1)$;\n2. Given $f(\\sqrt{x}-1)=x+2\\sqrt{x}$, find $f(x)$;\n3. Given $f(x)-2f(\\frac{1}{x})=3x+2$, find $f(x)$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 $f(x)$ be a periodic odd function defined on $\\mathbb{R}$, with the smallest positive period being $3$, and it satisfies $f(1) > -2$ and $f(2) = m - \\frac{3}{m}$. Determine the range of values for $m$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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: $({\\frac{π}{\\sqrt{3}-1}})^{0}-cos^{2}30°=$____.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 $Q$ be the plane passing through the point $(1, 1, 1)$ with normal vector $\\begin{pmatrix} 2 \\\\ -1 \\\\ 1 \\end{pmatrix}$. Find the matrix $\\mathbf{S}$ such that for any vector $\\mathbf{u}$, $\\mathbf{S} \\mathbf{u}$ is the reflection of $\\mathbf{u}$ through plane $Q$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Thirty-five teams play a tournament where each team plays against every other team exactly once, and there are no ties allowed. Each game has a $50\\%$ chance of being won by either team. Find the probability that no two teams will end up with the same number of victories in a simplified form $\\frac{m}{n}$ where $m$ and $n$ are coprime, and compute $\\log_2 n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 the function \\( f(x) = \\frac{b}{3x - 4} \\). If \\( f(3) = f^{-1}(b + 2) \\), find the sum of all possible values of \\( b \\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Katie will roll two six-sided dice. What is the probability that she will roll a number less than five on the first die and an even 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" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 6-digit number is called flippy if its digits alternate between two distinct digits. How many 6-digit flippy numbers are divisible by 40?\nA) 1\nB) 2\nC) 3\nD) 4\nE) 5\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Amelia has a coin that lands heads with probability $\\frac{1}{4}$, and Blaine has a coin that lands on heads with probability $\\frac{1}{3}$. They alternately toss their coins, starting with Amelia, until someone gets a head or they both toss their coins 5 times without a head, resulting in a tie. All coin tosses are independent. Determine the probability that Amelia wins.\nA) $\\frac{1}{4}$\nB) $\\frac{15}{32}$\nC) $\\frac{1}{2}$\nD) $\\frac{1}{3}$\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 pyramid with a square base $ABCD$ where $AB = 10$ and height $20$ is cut by a plane parallel to its base, creating a smaller pyramid and a frustum. Let $X$ be the center of the circumsphere of the frustum and $T$ the apex of the pyramid. If the volume of the original pyramid is nine times that of the smaller pyramid, find the distance $XT$ as a simplified fraction and compute the sum of its numerator and denominator.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Among the years 1700, 1996, 1998, and 2036, which are leap years and which are common years?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 halfway between $\\frac{1}{4}$ and $\\frac{1}{7}$ is?\nA) $\\frac{11}{25}$\nB) $\\frac{11}{56}$ \nC) $\\frac{25}{56}$\nD) $\\frac{3}{28}$\nE) $\\frac{3}{56}$\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "We have an octahedron with 5 blue faces and 3 red faces. What is the probability that when it is rolled, a blue face will be facing up?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 $f(x)=x^{2}+a\\ln x$ $(a\\in\\mathbb{R})$,\n\n$(1)$ If the tangent line of the function $f(x)$ at $x=1$ is perpendicular to the $y$-axis, find the value of $a$;\n\n$(2)$ If the function $f(x)$ is increasing on the interval $(1,+\\infty)$, find the range of $a$;\n\n$(3)$ Discuss the monotonicity of the function $g(x)=f(x)-(a+2)x$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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: $(-1) \\times (-4) + 2^{2} \\div (7-5)$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 sequence \\\\(\\{a_n\\}\\\\) satisfies \\\\(a_{n+1}+a_n=n\\\\), if \\\\(a_1=2\\\\), find \\\\(a_4-a_2\\\\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "At the \"Lukomorye\" station, tickets are sold for one, five, and twenty trips. All tickets cost an integer number of gold coins. Five single-trip tickets are more expensive than one five-trip ticket, and four five-trip tickets are more expensive than one twenty-trip ticket. It turned out that the cheapest way for 33 warriors to travel was to buy tickets for 35 trips, spending 33 gold coins. How much does a five-trip ticket cost?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 $i(3+4i)$, evaluate the expression.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "BoatWorks built 4 canoes in January of this year and then each subsequent calendar month they built three times the number of canoes they had built the previous month. How many total canoes were built by BoatWorks by the end of April of this year?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Given sets $A=\\{x| -1 \\lt x \\lt 1\\}$ and $B=\\{x| 0\\leqslant x\\leqslant 2\\}$, 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" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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\n\\[2^{-\\frac{5}{2} + 3 \\cos \\theta} + 1 = 2^{\\frac{1}{2} + \\cos \\theta},\\]\ncompute $\\cos 2 \\theta.$\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 proposition $p$: \"For all $x \\in [1,2]$, $x^{2}-a \\geqslant 0$\"; proposition $q$: \"There exists $x \\in \\mathbb{R}$, such that $x^{2}+2ax+2-a=0$\", if the proposition \"$p \\land q$\" is true, then the range of the real number $a$ is \\_\\_\\_\\_\\_\\_.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 derivative of the function $f(x)$ satisfies $f'(3)=9$, find the value of the derivative of $f(3x^{2})$ at $x=1$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 diagonals of a rhombus measure 20 feet and 16 feet. What is the perimeter of the rhombus? Express your answer in simplest radical form.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Given the equation with respect to $x$, $\\cos^2(x+\\pi) - \\sin x + a = 0$,\n(1) If $x = \\frac{5\\pi}{6}$ is a solution of this equation, find the value of $a$;\n(2) If this equation has a solution, determine 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" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 $A = \\{1, 2, 4\\}$ and $B = \\{m, 4, 7\\}$. If $A \\cap B = \\{1, 4\\}$, then $A \\cup B = \\_\\_\\_\\_\\_\\_$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 $x < \\frac{1}{2}$, find the maximum value of the function $y=\\frac{1}{4x-2}+4x-5$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 vector $\\overrightarrow{a}=(1,\\lambda,2)$, $\\overrightarrow{b}=(2,-1,2)$, and $\\overrightarrow{a}\\perp\\overrightarrow{b}$, then $\\lambda$ equals \\_\\_\\_\\_\\_\\_\\_\\_.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 $3x^{m+n} - 5y^{n-m} = 0$ is a linear equation in two variables, then $m=$____, $n=$____.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "In triangle $ABC$, $BC = 40$ and $\\angle C = 45^\\circ$. Let the perpendicular bisector of $BC$ intersect $BC$ and $AC$ at $D$ and $E$, respectively. Find the length of $DE$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 three numbers \\(x\\), \\(y\\), and \\(z\\) is 150. If we increase \\(x\\) by 10, decrease \\(y\\) by 10, and multiply \\(z\\) by 3, the three resulting numbers are equal. 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" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 all real numbers $x$, evaluate the expression $x[x\\{x(3-x) - 5\\} + 12] + 2 =$\nA) $-x^4 + x^3 - 5x^2 + 12x + 2$\nB) $-x^4 + 3x^3 - 5x^2 + 10x + 2$\nC) $-x^4 + 3x^3 - 5x^2 + 12x + 2$\nD) $x^4 + 3x^3 - 5x^2 + 12x + 2$\nE) $-x^4 - 3x^3 + 5x^2 - 12x - 2$\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 quadratic $2x^2 + 5x + 3$ can be rewritten in the form $3(x - 1)^2 + k$. When the quadratic $4(2x^2 + 5x + 3)$ is expressed in the form $m(x - h)^2 + p$, what is the value of $h$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 the coordinate plane $(x, y)$, a circle with a radius of 4 and centered at the origin is drawn. A line given by the equation $y = 4 - (2 - \\sqrt{3}) x$ intersects the circle at points $A$ and $B$. Find the sum of the length of the segment $A B$ and the length of the shorter arc $A B$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\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 the following equations: \n$(1) 2x^{2}-2=x$; \n$(2) x(x-2)+x-2=0$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Suppose that at the end of one year, one unit of money has depreciated by $10\\%$ compared to the beginning of that year. Find the smallest integer $n$ such that after $n$ years, one unit of money has depreciated by at least $90\\%$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]

[ { "content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n", "role": "system" }, { "content": "Ella walks to her university library, averaging 80 steps per minute, with each of her steps covering 80 cm. It takes her 20 minutes to get to the library. Her friend Tia, going to the same library by the same route, averages 120 steps per minute, but her steps are only 70 cm long. Calculate the time it takes Tia to reach the library in minutes.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user" }, { "content": "I am a dummy message", "role": "user" } ]