Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. $\frac{p}{q}=\cos(\angle CMD)$ is irreducible fraction, what is the value of $p+q$?
Response:
Step 1: Without loss of generality, let the edge-length of $ABCD$ be $2.$ It follows that $MC=MD=\sqrt3.$
Step 2: Let $O$ be the center of $\triangle ABD,$ so $\overline{CO}\perp\overline{MOD}.$ Note that $MO=\frac13 MD=\frac{\sqrt{3}}{3}.$
Step 3: In right $\triangle CMO,$ we have\[\cos(\angle CMD)=\frac{MO}{MC}=\frac13.\]
Step 4: So the answer is $1+3=4.$
Final Answer: \boxed{4} ####

Question: How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser numbers?
Response:
Step 1: Clearly, the integers from $8$ through $14$ must be in different pairs, so are the integers from $1$ through $7.$ Note that $7$ must pair with $14.$
Step 2: We pair the numbers $1,2,3,4,5,6$ with the numbers $8,9,10,11,12,13$ systematically.
Step 3: $6$ can pair with either $12$ or $13.$
Step 4: $5$ can pair with any of the three remaining numbers from $10,11,12,13.$
Step 5: $1,2,3,4$ can pair with the other four remaining numbers from $8,9,10,11,12,13$ without restrictions.
Step 6: Together, the answer is $2\cdot3\cdot4!=144.$
Final Answer: \boxed{144} ####

Question: The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box?
Response:
Step 1: Let $P(x) = 10x^3 - 39x^2 + 29x - 6$, and let $a, b, c$ be the roots of $P(x)$.
Step 2: The roots of $P(x-2)$ are then $a + 2, b + 2, c + 2,$ so the product of the roots of $P(x-2)$ is the area of the desired rectangular prism.
Step 3: $P(x-2)$ has leading coefficient $10$ and constant term $P(0-2) = P(-2) = 10(-2)^3 - 39(-2)^2 + 29(-2) - 6 = -300$.
Step 4: Thus, by Vieta's Formulas, the product of the roots of $P(x-2)$ is $\frac{-(-300)}{10} = 30$.
Final Answer: \boxed{30} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: What is the value of $9901 \cdot 101 - 99 \cdot 10101$?
Response:

Output: 2


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. $\frac{p}{q}=\cos(\angle CMD)$ is irreducible fraction, what is the value of $p+q$?
Response:
Step 1: Without loss of generality, let the edge-length of $ABCD$ be $2.$ It follows that $MC=MD=\sqrt3.$
Step 2: Let $O$ be the center of $\triangle ABD,$ so $\overline{CO}\perp\overline{MOD}.$ Note that $MO=\frac13 MD=\frac{\sqrt{3}}{3}.$
Step 3: In right $\triangle CMO,$ we have\[\cos(\angle CMD)=\frac{MO}{MC}=\frac13.\]
Step 4: So the answer is $1+3=4.$
Final Answer: \boxed{4} ####

Question: How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser numbers?
Response:
Step 1: Clearly, the integers from $8$ through $14$ must be in different pairs, so are the integers from $1$ through $7.$ Note that $7$ must pair with $14.$
Step 2: We pair the numbers $1,2,3,4,5,6$ with the numbers $8,9,10,11,12,13$ systematically.
Step 3: $6$ can pair with either $12$ or $13.$
Step 4: $5$ can pair with any of the three remaining numbers from $10,11,12,13.$
Step 5: $1,2,3,4$ can pair with the other four remaining numbers from $8,9,10,11,12,13$ without restrictions.
Step 6: Together, the answer is $2\cdot3\cdot4!=144.$
Final Answer: \boxed{144} ####

Question: The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box?
Response:
Step 1: Let $P(x) = 10x^3 - 39x^2 + 29x - 6$, and let $a, b, c$ be the roots of $P(x)$.
Step 2: The roots of $P(x-2)$ are then $a + 2, b + 2, c + 2,$ so the product of the roots of $P(x-2)$ is the area of the desired rectangular prism.
Step 3: $P(x-2)$ has leading coefficient $10$ and constant term $P(0-2) = P(-2) = 10(-2)^3 - 39(-2)^2 + 29(-2) - 6 = -300$.
Step 4: Thus, by Vieta's Formulas, the product of the roots of $P(x-2)$ is $\frac{-(-300)}{10} = 30$.
Final Answer: \boxed{30} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: A model used to estimate the time it will take to hike to the top of the mountain on a trail is of the form $T = aL + bG$, where $a$ and $b$ are constants, $T$ is the time in minutes, $L$ is the length of the trail in miles, and $G$ is the altitude gain in feet. The model estimates that it will take $69$ minutes to hike to the top if a trail is $1.5$ miles long and ascends $800$ feet, as well as if a trail is $1.2$ miles long and ascends $1100$ feet. How many minutes does the model estimates it will take to hike to the top if the trail is $4.2$ miles long and ascends $4000$ feet?
Response:

Output: 246


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. $\frac{p}{q}=\cos(\angle CMD)$ is irreducible fraction, what is the value of $p+q$?
Response:
Step 1: Without loss of generality, let the edge-length of $ABCD$ be $2.$ It follows that $MC=MD=\sqrt3.$
Step 2: Let $O$ be the center of $\triangle ABD,$ so $\overline{CO}\perp\overline{MOD}.$ Note that $MO=\frac13 MD=\frac{\sqrt{3}}{3}.$
Step 3: In right $\triangle CMO,$ we have\[\cos(\angle CMD)=\frac{MO}{MC}=\frac13.\]
Step 4: So the answer is $1+3=4.$
Final Answer: \boxed{4} ####

Question: How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser numbers?
Response:
Step 1: Clearly, the integers from $8$ through $14$ must be in different pairs, so are the integers from $1$ through $7.$ Note that $7$ must pair with $14.$
Step 2: We pair the numbers $1,2,3,4,5,6$ with the numbers $8,9,10,11,12,13$ systematically.
Step 3: $6$ can pair with either $12$ or $13.$
Step 4: $5$ can pair with any of the three remaining numbers from $10,11,12,13.$
Step 5: $1,2,3,4$ can pair with the other four remaining numbers from $8,9,10,11,12,13$ without restrictions.
Step 6: Together, the answer is $2\cdot3\cdot4!=144.$
Final Answer: \boxed{144} ####

Question: The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box?
Response:
Step 1: Let $P(x) = 10x^3 - 39x^2 + 29x - 6$, and let $a, b, c$ be the roots of $P(x)$.
Step 2: The roots of $P(x-2)$ are then $a + 2, b + 2, c + 2,$ so the product of the roots of $P(x-2)$ is the area of the desired rectangular prism.
Step 3: $P(x-2)$ has leading coefficient $10$ and constant term $P(0-2) = P(-2) = 10(-2)^3 - 39(-2)^2 + 29(-2) - 6 = -300$.
Step 4: Thus, by Vieta's Formulas, the product of the roots of $P(x-2)$ is $\frac{-(-300)}{10} = 30$.
Final Answer: \boxed{30} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: The number $2024$ is written as the sum of not necessarily distinct two-digit numbers. What is the least number of two-digit numbers needed to write this sum?
Response:

Output: 21


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. $\frac{p}{q}=\cos(\angle CMD)$ is irreducible fraction, what is the value of $p+q$?
Response:
Step 1: Without loss of generality, let the edge-length of $ABCD$ be $2.$ It follows that $MC=MD=\sqrt3.$
Step 2: Let $O$ be the center of $\triangle ABD,$ so $\overline{CO}\perp\overline{MOD}.$ Note that $MO=\frac13 MD=\frac{\sqrt{3}}{3}.$
Step 3: In right $\triangle CMO,$ we have\[\cos(\angle CMD)=\frac{MO}{MC}=\frac13.\]
Step 4: So the answer is $1+3=4.$
Final Answer: \boxed{4} ####

Question: How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser numbers?
Response:
Step 1: Clearly, the integers from $8$ through $14$ must be in different pairs, so are the integers from $1$ through $7.$ Note that $7$ must pair with $14.$
Step 2: We pair the numbers $1,2,3,4,5,6$ with the numbers $8,9,10,11,12,13$ systematically.
Step 3: $6$ can pair with either $12$ or $13.$
Step 4: $5$ can pair with any of the three remaining numbers from $10,11,12,13.$
Step 5: $1,2,3,4$ can pair with the other four remaining numbers from $8,9,10,11,12,13$ without restrictions.
Step 6: Together, the answer is $2\cdot3\cdot4!=144.$
Final Answer: \boxed{144} ####

Question: The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box?
Response:
Step 1: Let $P(x) = 10x^3 - 39x^2 + 29x - 6$, and let $a, b, c$ be the roots of $P(x)$.
Step 2: The roots of $P(x-2)$ are then $a + 2, b + 2, c + 2,$ so the product of the roots of $P(x-2)$ is the area of the desired rectangular prism.
Step 3: $P(x-2)$ has leading coefficient $10$ and constant term $P(0-2) = P(-2) = 10(-2)^3 - 39(-2)^2 + 29(-2) - 6 = -300$.
Step 4: Thus, by Vieta's Formulas, the product of the roots of $P(x-2)$ is $\frac{-(-300)}{10} = 30$.
Final Answer: \boxed{30} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: What is the least value of $n$ such that $n!$ is a multiple of $2024$?
Response:

Output: 23


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. $\frac{p}{q}=\cos(\angle CMD)$ is irreducible fraction, what is the value of $p+q$?
Response:
Step 1: Without loss of generality, let the edge-length of $ABCD$ be $2.$ It follows that $MC=MD=\sqrt3.$
Step 2: Let $O$ be the center of $\triangle ABD,$ so $\overline{CO}\perp\overline{MOD}.$ Note that $MO=\frac13 MD=\frac{\sqrt{3}}{3}.$
Step 3: In right $\triangle CMO,$ we have\[\cos(\angle CMD)=\frac{MO}{MC}=\frac13.\]
Step 4: So the answer is $1+3=4.$
Final Answer: \boxed{4} ####

Question: How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser numbers?
Response:
Step 1: Clearly, the integers from $8$ through $14$ must be in different pairs, so are the integers from $1$ through $7.$ Note that $7$ must pair with $14.$
Step 2: We pair the numbers $1,2,3,4,5,6$ with the numbers $8,9,10,11,12,13$ systematically.
Step 3: $6$ can pair with either $12$ or $13.$
Step 4: $5$ can pair with any of the three remaining numbers from $10,11,12,13.$
Step 5: $1,2,3,4$ can pair with the other four remaining numbers from $8,9,10,11,12,13$ without restrictions.
Step 6: Together, the answer is $2\cdot3\cdot4!=144.$
Final Answer: \boxed{144} ####

Question: The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box?
Response:
Step 1: Let $P(x) = 10x^3 - 39x^2 + 29x - 6$, and let $a, b, c$ be the roots of $P(x)$.
Step 2: The roots of $P(x-2)$ are then $a + 2, b + 2, c + 2,$ so the product of the roots of $P(x-2)$ is the area of the desired rectangular prism.
Step 3: $P(x-2)$ has leading coefficient $10$ and constant term $P(0-2) = P(-2) = 10(-2)^3 - 39(-2)^2 + 29(-2) - 6 = -300$.
Step 4: Thus, by Vieta's Formulas, the product of the roots of $P(x-2)$ is $\frac{-(-300)}{10} = 30$.
Final Answer: \boxed{30} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: A data set containing $20$ numbers, some of which are $6$, has mean $45$. When all the 6s are removed, the data set has mean $66$. How many 6s were in the original data set?
Response:

Output: 7


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. $\frac{p}{q}=\cos(\angle CMD)$ is irreducible fraction, what is the value of $p+q$?
Response:
Step 1: Without loss of generality, let the edge-length of $ABCD$ be $2.$ It follows that $MC=MD=\sqrt3.$
Step 2: Let $O$ be the center of $\triangle ABD,$ so $\overline{CO}\perp\overline{MOD}.$ Note that $MO=\frac13 MD=\frac{\sqrt{3}}{3}.$
Step 3: In right $\triangle CMO,$ we have\[\cos(\angle CMD)=\frac{MO}{MC}=\frac13.\]
Step 4: So the answer is $1+3=4.$
Final Answer: \boxed{4} ####

Question: How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser numbers?
Response:
Step 1: Clearly, the integers from $8$ through $14$ must be in different pairs, so are the integers from $1$ through $7.$ Note that $7$ must pair with $14.$
Step 2: We pair the numbers $1,2,3,4,5,6$ with the numbers $8,9,10,11,12,13$ systematically.
Step 3: $6$ can pair with either $12$ or $13.$
Step 4: $5$ can pair with any of the three remaining numbers from $10,11,12,13.$
Step 5: $1,2,3,4$ can pair with the other four remaining numbers from $8,9,10,11,12,13$ without restrictions.
Step 6: Together, the answer is $2\cdot3\cdot4!=144.$
Final Answer: \boxed{144} ####

Question: The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box?
Response:
Step 1: Let $P(x) = 10x^3 - 39x^2 + 29x - 6$, and let $a, b, c$ be the roots of $P(x)$.
Step 2: The roots of $P(x-2)$ are then $a + 2, b + 2, c + 2,$ so the product of the roots of $P(x-2)$ is the area of the desired rectangular prism.
Step 3: $P(x-2)$ has leading coefficient $10$ and constant term $P(0-2) = P(-2) = 10(-2)^3 - 39(-2)^2 + 29(-2) - 6 = -300$.
Step 4: Thus, by Vieta's Formulas, the product of the roots of $P(x-2)$ is $\frac{-(-300)}{10} = 30$.
Final Answer: \boxed{30} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: The product of three integers is $60$. What is the least possible positive sum of the three integers?
Response:

Output: 3


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. $\frac{p}{q}=\cos(\angle CMD)$ is irreducible fraction, what is the value of $p+q$?
Response:
Step 1: Without loss of generality, let the edge-length of $ABCD$ be $2.$ It follows that $MC=MD=\sqrt3.$
Step 2: Let $O$ be the center of $\triangle ABD,$ so $\overline{CO}\perp\overline{MOD}.$ Note that $MO=\frac13 MD=\frac{\sqrt{3}}{3}.$
Step 3: In right $\triangle CMO,$ we have\[\cos(\angle CMD)=\frac{MO}{MC}=\frac13.\]
Step 4: So the answer is $1+3=4.$
Final Answer: \boxed{4} ####

Question: How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser numbers?
Response:
Step 1: Clearly, the integers from $8$ through $14$ must be in different pairs, so are the integers from $1$ through $7.$ Note that $7$ must pair with $14.$
Step 2: We pair the numbers $1,2,3,4,5,6$ with the numbers $8,9,10,11,12,13$ systematically.
Step 3: $6$ can pair with either $12$ or $13.$
Step 4: $5$ can pair with any of the three remaining numbers from $10,11,12,13.$
Step 5: $1,2,3,4$ can pair with the other four remaining numbers from $8,9,10,11,12,13$ without restrictions.
Step 6: Together, the answer is $2\cdot3\cdot4!=144.$
Final Answer: \boxed{144} ####

Question: The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box?
Response:
Step 1: Let $P(x) = 10x^3 - 39x^2 + 29x - 6$, and let $a, b, c$ be the roots of $P(x)$.
Step 2: The roots of $P(x-2)$ are then $a + 2, b + 2, c + 2,$ so the product of the roots of $P(x-2)$ is the area of the desired rectangular prism.
Step 3: $P(x-2)$ has leading coefficient $10$ and constant term $P(0-2) = P(-2) = 10(-2)^3 - 39(-2)^2 + 29(-2) - 6 = -300$.
Step 4: Thus, by Vieta's Formulas, the product of the roots of $P(x-2)$ is $\frac{-(-300)}{10} = 30$.
Final Answer: \boxed{30} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: In $\triangle ABC$, $\angle ABC = 90^\circ$ and $BA = BC = \sqrt{2}$. Points $P_1, P_2, \ldots, P_{2024}$ lie on hypotenuse $\overline{AC}$ so that $AP_1 = P_1P_2 = P_2P_3 = \cdots = P_{2023}P_{2024} = P_{2024}C$. What is the length of the vector sum $\overrightarrow{BP_1} + \overrightarrow{BP_2} + \overrightarrow{BP_3} + \cdots + \overrightarrow{BP_{2024}}$?
Response:

Output: 2024


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. $\frac{p}{q}=\cos(\angle CMD)$ is irreducible fraction, what is the value of $p+q$?
Response:
Step 1: Without loss of generality, let the edge-length of $ABCD$ be $2.$ It follows that $MC=MD=\sqrt3.$
Step 2: Let $O$ be the center of $\triangle ABD,$ so $\overline{CO}\perp\overline{MOD}.$ Note that $MO=\frac13 MD=\frac{\sqrt{3}}{3}.$
Step 3: In right $\triangle CMO,$ we have\[\cos(\angle CMD)=\frac{MO}{MC}=\frac13.\]
Step 4: So the answer is $1+3=4.$
Final Answer: \boxed{4} ####

Question: How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser numbers?
Response:
Step 1: Clearly, the integers from $8$ through $14$ must be in different pairs, so are the integers from $1$ through $7.$ Note that $7$ must pair with $14.$
Step 2: We pair the numbers $1,2,3,4,5,6$ with the numbers $8,9,10,11,12,13$ systematically.
Step 3: $6$ can pair with either $12$ or $13.$
Step 4: $5$ can pair with any of the three remaining numbers from $10,11,12,13.$
Step 5: $1,2,3,4$ can pair with the other four remaining numbers from $8,9,10,11,12,13$ without restrictions.
Step 6: Together, the answer is $2\cdot3\cdot4!=144.$
Final Answer: \boxed{144} ####

Question: The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box?
Response:
Step 1: Let $P(x) = 10x^3 - 39x^2 + 29x - 6$, and let $a, b, c$ be the roots of $P(x)$.
Step 2: The roots of $P(x-2)$ are then $a + 2, b + 2, c + 2,$ so the product of the roots of $P(x-2)$ is the area of the desired rectangular prism.
Step 3: $P(x-2)$ has leading coefficient $10$ and constant term $P(0-2) = P(-2) = 10(-2)^3 - 39(-2)^2 + 29(-2) - 6 = -300$.
Step 4: Thus, by Vieta's Formulas, the product of the roots of $P(x-2)$ is $\frac{-(-300)}{10} = 30$.
Final Answer: \boxed{30} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: How many angles $\theta$ with $0 \leq \theta \leq 2\pi$ satisfy $\log(\sin(3\theta)) + \log(\cos(2\theta)) = 0$?
Response:

Output: 0


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. $\frac{p}{q}=\cos(\angle CMD)$ is irreducible fraction, what is the value of $p+q$?
Response:
Step 1: Without loss of generality, let the edge-length of $ABCD$ be $2.$ It follows that $MC=MD=\sqrt3.$
Step 2: Let $O$ be the center of $\triangle ABD,$ so $\overline{CO}\perp\overline{MOD}.$ Note that $MO=\frac13 MD=\frac{\sqrt{3}}{3}.$
Step 3: In right $\triangle CMO,$ we have\[\cos(\angle CMD)=\frac{MO}{MC}=\frac13.\]
Step 4: So the answer is $1+3=4.$
Final Answer: \boxed{4} ####

Question: How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser numbers?
Response:
Step 1: Clearly, the integers from $8$ through $14$ must be in different pairs, so are the integers from $1$ through $7.$ Note that $7$ must pair with $14.$
Step 2: We pair the numbers $1,2,3,4,5,6$ with the numbers $8,9,10,11,12,13$ systematically.
Step 3: $6$ can pair with either $12$ or $13.$
Step 4: $5$ can pair with any of the three remaining numbers from $10,11,12,13.$
Step 5: $1,2,3,4$ can pair with the other four remaining numbers from $8,9,10,11,12,13$ without restrictions.
Step 6: Together, the answer is $2\cdot3\cdot4!=144.$
Final Answer: \boxed{144} ####

Question: The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box?
Response:
Step 1: Let $P(x) = 10x^3 - 39x^2 + 29x - 6$, and let $a, b, c$ be the roots of $P(x)$.
Step 2: The roots of $P(x-2)$ are then $a + 2, b + 2, c + 2,$ so the product of the roots of $P(x-2)$ is the area of the desired rectangular prism.
Step 3: $P(x-2)$ has leading coefficient $10$ and constant term $P(0-2) = P(-2) = 10(-2)^3 - 39(-2)^2 + 29(-2) - 6 = -300$.
Step 4: Thus, by Vieta's Formulas, the product of the roots of $P(x-2)$ is $\frac{-(-300)}{10} = 30$.
Final Answer: \boxed{30} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: Let $M$ be the greatest integer such that both $M + 1213$ and $M + 3773$ are perfect squares. What is the units digit of $M$?
Response:

Output: 8


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. $\frac{p}{q}=\cos(\angle CMD)$ is irreducible fraction, what is the value of $p+q$?
Response:
Step 1: Without loss of generality, let the edge-length of $ABCD$ be $2.$ It follows that $MC=MD=\sqrt3.$
Step 2: Let $O$ be the center of $\triangle ABD,$ so $\overline{CO}\perp\overline{MOD}.$ Note that $MO=\frac13 MD=\frac{\sqrt{3}}{3}.$
Step 3: In right $\triangle CMO,$ we have\[\cos(\angle CMD)=\frac{MO}{MC}=\frac13.\]
Step 4: So the answer is $1+3=4.$
Final Answer: \boxed{4} ####

Question: How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser numbers?
Response:
Step 1: Clearly, the integers from $8$ through $14$ must be in different pairs, so are the integers from $1$ through $7.$ Note that $7$ must pair with $14.$
Step 2: We pair the numbers $1,2,3,4,5,6$ with the numbers $8,9,10,11,12,13$ systematically.
Step 3: $6$ can pair with either $12$ or $13.$
Step 4: $5$ can pair with any of the three remaining numbers from $10,11,12,13.$
Step 5: $1,2,3,4$ can pair with the other four remaining numbers from $8,9,10,11,12,13$ without restrictions.
Step 6: Together, the answer is $2\cdot3\cdot4!=144.$
Final Answer: \boxed{144} ####

Question: The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box?
Response:
Step 1: Let $P(x) = 10x^3 - 39x^2 + 29x - 6$, and let $a, b, c$ be the roots of $P(x)$.
Step 2: The roots of $P(x-2)$ are then $a + 2, b + 2, c + 2,$ so the product of the roots of $P(x-2)$ is the area of the desired rectangular prism.
Step 3: $P(x-2)$ has leading coefficient $10$ and constant term $P(0-2) = P(-2) = 10(-2)^3 - 39(-2)^2 + 29(-2) - 6 = -300$.
Step 4: Thus, by Vieta's Formulas, the product of the roots of $P(x-2)$ is $\frac{-(-300)}{10} = 30$.
Final Answer: \boxed{30} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: Let $\alpha$ be the radian measure of the smallest angle in a $3-4-5$ right triangle. Let $\beta$ be the radian measure of the smallest angle in a $7-24-25$ right triangle. In terms of $\alpha$, what is $\beta$?
Response:

Output: \frac{\pi}{2} - 2\alpha


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. $\frac{p}{q}=\cos(\angle CMD)$ is irreducible fraction, what is the value of $p+q$?
Response:
Step 1: Without loss of generality, let the edge-length of $ABCD$ be $2.$ It follows that $MC=MD=\sqrt3.$
Step 2: Let $O$ be the center of $\triangle ABD,$ so $\overline{CO}\perp\overline{MOD}.$ Note that $MO=\frac13 MD=\frac{\sqrt{3}}{3}.$
Step 3: In right $\triangle CMO,$ we have\[\cos(\angle CMD)=\frac{MO}{MC}=\frac13.\]
Step 4: So the answer is $1+3=4.$
Final Answer: \boxed{4} ####

Question: How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser numbers?
Response:
Step 1: Clearly, the integers from $8$ through $14$ must be in different pairs, so are the integers from $1$ through $7.$ Note that $7$ must pair with $14.$
Step 2: We pair the numbers $1,2,3,4,5,6$ with the numbers $8,9,10,11,12,13$ systematically.
Step 3: $6$ can pair with either $12$ or $13.$
Step 4: $5$ can pair with any of the three remaining numbers from $10,11,12,13.$
Step 5: $1,2,3,4$ can pair with the other four remaining numbers from $8,9,10,11,12,13$ without restrictions.
Step 6: Together, the answer is $2\cdot3\cdot4!=144.$
Final Answer: \boxed{144} ####

Question: The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box?
Response:
Step 1: Let $P(x) = 10x^3 - 39x^2 + 29x - 6$, and let $a, b, c$ be the roots of $P(x)$.
Step 2: The roots of $P(x-2)$ are then $a + 2, b + 2, c + 2,$ so the product of the roots of $P(x-2)$ is the area of the desired rectangular prism.
Step 3: $P(x-2)$ has leading coefficient $10$ and constant term $P(0-2) = P(-2) = 10(-2)^3 - 39(-2)^2 + 29(-2) - 6 = -300$.
Step 4: Thus, by Vieta's Formulas, the product of the roots of $P(x-2)$ is $\frac{-(-300)}{10} = 30$.
Final Answer: \boxed{30} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: There are exactly $K$ positive integers $b$ with $5 \leq b \leq 2024$ such that the base-$b$ integer $2024_b$ is divisible by $16$ (where $16$ is in base ten). What is the sum of the digits of $K$?
Response:

Output: 20


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. $\frac{p}{q}=\cos(\angle CMD)$ is irreducible fraction, what is the value of $p+q$?
Response:
Step 1: Without loss of generality, let the edge-length of $ABCD$ be $2.$ It follows that $MC=MD=\sqrt3.$
Step 2: Let $O$ be the center of $\triangle ABD,$ so $\overline{CO}\perp\overline{MOD}.$ Note that $MO=\frac13 MD=\frac{\sqrt{3}}{3}.$
Step 3: In right $\triangle CMO,$ we have\[\cos(\angle CMD)=\frac{MO}{MC}=\frac13.\]
Step 4: So the answer is $1+3=4.$
Final Answer: \boxed{4} ####

Question: How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser numbers?
Response:
Step 1: Clearly, the integers from $8$ through $14$ must be in different pairs, so are the integers from $1$ through $7.$ Note that $7$ must pair with $14.$
Step 2: We pair the numbers $1,2,3,4,5,6$ with the numbers $8,9,10,11,12,13$ systematically.
Step 3: $6$ can pair with either $12$ or $13.$
Step 4: $5$ can pair with any of the three remaining numbers from $10,11,12,13.$
Step 5: $1,2,3,4$ can pair with the other four remaining numbers from $8,9,10,11,12,13$ without restrictions.
Step 6: Together, the answer is $2\cdot3\cdot4!=144.$
Final Answer: \boxed{144} ####

Question: The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box?
Response:
Step 1: Let $P(x) = 10x^3 - 39x^2 + 29x - 6$, and let $a, b, c$ be the roots of $P(x)$.
Step 2: The roots of $P(x-2)$ are then $a + 2, b + 2, c + 2,$ so the product of the roots of $P(x-2)$ is the area of the desired rectangular prism.
Step 3: $P(x-2)$ has leading coefficient $10$ and constant term $P(0-2) = P(-2) = 10(-2)^3 - 39(-2)^2 + 29(-2) - 6 = -300$.
Step 4: Thus, by Vieta's Formulas, the product of the roots of $P(x-2)$ is $\frac{-(-300)}{10} = 30$.
Final Answer: \boxed{30} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: The first three terms of a geometric sequence are the integers $a$, $720$, and $b$, where $a < 720 < b$. What is the sum of the digits of the least possible value of $b$?
Response:

Output: 21


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. $\frac{p}{q}=\cos(\angle CMD)$ is irreducible fraction, what is the value of $p+q$?
Response:
Step 1: Without loss of generality, let the edge-length of $ABCD$ be $2.$ It follows that $MC=MD=\sqrt3.$
Step 2: Let $O$ be the center of $\triangle ABD,$ so $\overline{CO}\perp\overline{MOD}.$ Note that $MO=\frac13 MD=\frac{\sqrt{3}}{3}.$
Step 3: In right $\triangle CMO,$ we have\[\cos(\angle CMD)=\frac{MO}{MC}=\frac13.\]
Step 4: So the answer is $1+3=4.$
Final Answer: \boxed{4} ####

Question: How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser numbers?
Response:
Step 1: Clearly, the integers from $8$ through $14$ must be in different pairs, so are the integers from $1$ through $7.$ Note that $7$ must pair with $14.$
Step 2: We pair the numbers $1,2,3,4,5,6$ with the numbers $8,9,10,11,12,13$ systematically.
Step 3: $6$ can pair with either $12$ or $13.$
Step 4: $5$ can pair with any of the three remaining numbers from $10,11,12,13.$
Step 5: $1,2,3,4$ can pair with the other four remaining numbers from $8,9,10,11,12,13$ without restrictions.
Step 6: Together, the answer is $2\cdot3\cdot4!=144.$
Final Answer: \boxed{144} ####

Question: The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box?
Response:
Step 1: Let $P(x) = 10x^3 - 39x^2 + 29x - 6$, and let $a, b, c$ be the roots of $P(x)$.
Step 2: The roots of $P(x-2)$ are then $a + 2, b + 2, c + 2,$ so the product of the roots of $P(x-2)$ is the area of the desired rectangular prism.
Step 3: $P(x-2)$ has leading coefficient $10$ and constant term $P(0-2) = P(-2) = 10(-2)^3 - 39(-2)^2 + 29(-2) - 6 = -300$.
Step 4: Thus, by Vieta's Formulas, the product of the roots of $P(x-2)$ is $\frac{-(-300)}{10} = 30$.
Final Answer: \boxed{30} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: The graph of $y = e^{x+1} + e^{-x} - 2$ has an axis of symmetry. What is the reflection of the point $(-1, \frac{1}{2})$ over this axis?
Response:

Output: (0, \frac{1}{2})


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. $\frac{p}{q}=\cos(\angle CMD)$ is irreducible fraction, what is the value of $p+q$?
Response:
Step 1: Without loss of generality, let the edge-length of $ABCD$ be $2.$ It follows that $MC=MD=\sqrt3.$
Step 2: Let $O$ be the center of $\triangle ABD,$ so $\overline{CO}\perp\overline{MOD}.$ Note that $MO=\frac13 MD=\frac{\sqrt{3}}{3}.$
Step 3: In right $\triangle CMO,$ we have\[\cos(\angle CMD)=\frac{MO}{MC}=\frac13.\]
Step 4: So the answer is $1+3=4.$
Final Answer: \boxed{4} ####

Question: How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser numbers?
Response:
Step 1: Clearly, the integers from $8$ through $14$ must be in different pairs, so are the integers from $1$ through $7.$ Note that $7$ must pair with $14.$
Step 2: We pair the numbers $1,2,3,4,5,6$ with the numbers $8,9,10,11,12,13$ systematically.
Step 3: $6$ can pair with either $12$ or $13.$
Step 4: $5$ can pair with any of the three remaining numbers from $10,11,12,13.$
Step 5: $1,2,3,4$ can pair with the other four remaining numbers from $8,9,10,11,12,13$ without restrictions.
Step 6: Together, the answer is $2\cdot3\cdot4!=144.$
Final Answer: \boxed{144} ####

Question: The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box?
Response:
Step 1: Let $P(x) = 10x^3 - 39x^2 + 29x - 6$, and let $a, b, c$ be the roots of $P(x)$.
Step 2: The roots of $P(x-2)$ are then $a + 2, b + 2, c + 2,$ so the product of the roots of $P(x-2)$ is the area of the desired rectangular prism.
Step 3: $P(x-2)$ has leading coefficient $10$ and constant term $P(0-2) = P(-2) = 10(-2)^3 - 39(-2)^2 + 29(-2) - 6 = -300$.
Step 4: Thus, by Vieta's Formulas, the product of the roots of $P(x-2)$ is $\frac{-(-300)}{10} = 30$.
Final Answer: \boxed{30} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: The roots of $x^3 + 2x^2 - x + 3$ are $p$, $q$, and $r$. What is the value of $(p^2 + 4)(q^2 + 4)(r^2 + 4)$?
Response:

Output: 125


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. $\frac{p}{q}=\cos(\angle CMD)$ is irreducible fraction, what is the value of $p+q$?
Response:
Step 1: Without loss of generality, let the edge-length of $ABCD$ be $2.$ It follows that $MC=MD=\sqrt3.$
Step 2: Let $O$ be the center of $\triangle ABD,$ so $\overline{CO}\perp\overline{MOD}.$ Note that $MO=\frac13 MD=\frac{\sqrt{3}}{3}.$
Step 3: In right $\triangle CMO,$ we have\[\cos(\angle CMD)=\frac{MO}{MC}=\frac13.\]
Step 4: So the answer is $1+3=4.$
Final Answer: \boxed{4} ####

Question: How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser numbers?
Response:
Step 1: Clearly, the integers from $8$ through $14$ must be in different pairs, so are the integers from $1$ through $7.$ Note that $7$ must pair with $14.$
Step 2: We pair the numbers $1,2,3,4,5,6$ with the numbers $8,9,10,11,12,13$ systematically.
Step 3: $6$ can pair with either $12$ or $13.$
Step 4: $5$ can pair with any of the three remaining numbers from $10,11,12,13.$
Step 5: $1,2,3,4$ can pair with the other four remaining numbers from $8,9,10,11,12,13$ without restrictions.
Step 6: Together, the answer is $2\cdot3\cdot4!=144.$
Final Answer: \boxed{144} ####

Question: The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box?
Response:
Step 1: Let $P(x) = 10x^3 - 39x^2 + 29x - 6$, and let $a, b, c$ be the roots of $P(x)$.
Step 2: The roots of $P(x-2)$ are then $a + 2, b + 2, c + 2,$ so the product of the roots of $P(x-2)$ is the area of the desired rectangular prism.
Step 3: $P(x-2)$ has leading coefficient $10$ and constant term $P(0-2) = P(-2) = 10(-2)^3 - 39(-2)^2 + 29(-2) - 6 = -300$.
Step 4: Thus, by Vieta's Formulas, the product of the roots of $P(x-2)$ is $\frac{-(-300)}{10} = 30$.
Final Answer: \boxed{30} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: A set of $12$ tokens — $3$ red, $2$ white, $1$ blue, and $6$ black — is to be distributed at random to $3$ game players, $4$ tokens per player. The probability that some player gets all the red tokens, another gets all the white tokens, and the remaining player gets the blue token can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$?
Response:

Output: 389


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. $\frac{p}{q}=\cos(\angle CMD)$ is irreducible fraction, what is the value of $p+q$?
Response:
Step 1: Without loss of generality, let the edge-length of $ABCD$ be $2.$ It follows that $MC=MD=\sqrt3.$
Step 2: Let $O$ be the center of $\triangle ABD,$ so $\overline{CO}\perp\overline{MOD}.$ Note that $MO=\frac13 MD=\frac{\sqrt{3}}{3}.$
Step 3: In right $\triangle CMO,$ we have\[\cos(\angle CMD)=\frac{MO}{MC}=\frac13.\]
Step 4: So the answer is $1+3=4.$
Final Answer: \boxed{4} ####

Question: How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser numbers?
Response:
Step 1: Clearly, the integers from $8$ through $14$ must be in different pairs, so are the integers from $1$ through $7.$ Note that $7$ must pair with $14.$
Step 2: We pair the numbers $1,2,3,4,5,6$ with the numbers $8,9,10,11,12,13$ systematically.
Step 3: $6$ can pair with either $12$ or $13.$
Step 4: $5$ can pair with any of the three remaining numbers from $10,11,12,13.$
Step 5: $1,2,3,4$ can pair with the other four remaining numbers from $8,9,10,11,12,13$ without restrictions.
Step 6: Together, the answer is $2\cdot3\cdot4!=144.$
Final Answer: \boxed{144} ####

Question: The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box?
Response:
Step 1: Let $P(x) = 10x^3 - 39x^2 + 29x - 6$, and let $a, b, c$ be the roots of $P(x)$.
Step 2: The roots of $P(x-2)$ are then $a + 2, b + 2, c + 2,$ so the product of the roots of $P(x-2)$ is the area of the desired rectangular prism.
Step 3: $P(x-2)$ has leading coefficient $10$ and constant term $P(0-2) = P(-2) = 10(-2)^3 - 39(-2)^2 + 29(-2) - 6 = -300$.
Step 4: Thus, by Vieta's Formulas, the product of the roots of $P(x-2)$ is $\frac{-(-300)}{10} = 30$.
Final Answer: \boxed{30} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: Integers $a$, $b$, and $c$ satisfy $ab + c = 100$, $bc + a = 87$, and $ca + b = 60$. What is $ab + bc + ca$?
Response:

Output: 276


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. $\frac{p}{q}=\cos(\angle CMD)$ is irreducible fraction, what is the value of $p+q$?
Response:
Step 1: Without loss of generality, let the edge-length of $ABCD$ be $2.$ It follows that $MC=MD=\sqrt3.$
Step 2: Let $O$ be the center of $\triangle ABD,$ so $\overline{CO}\perp\overline{MOD}.$ Note that $MO=\frac13 MD=\frac{\sqrt{3}}{3}.$
Step 3: In right $\triangle CMO,$ we have\[\cos(\angle CMD)=\frac{MO}{MC}=\frac13.\]
Step 4: So the answer is $1+3=4.$
Final Answer: \boxed{4} ####

Question: How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser numbers?
Response:
Step 1: Clearly, the integers from $8$ through $14$ must be in different pairs, so are the integers from $1$ through $7.$ Note that $7$ must pair with $14.$
Step 2: We pair the numbers $1,2,3,4,5,6$ with the numbers $8,9,10,11,12,13$ systematically.
Step 3: $6$ can pair with either $12$ or $13.$
Step 4: $5$ can pair with any of the three remaining numbers from $10,11,12,13.$
Step 5: $1,2,3,4$ can pair with the other four remaining numbers from $8,9,10,11,12,13$ without restrictions.
Step 6: Together, the answer is $2\cdot3\cdot4!=144.$
Final Answer: \boxed{144} ####

Question: The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box?
Response:
Step 1: Let $P(x) = 10x^3 - 39x^2 + 29x - 6$, and let $a, b, c$ be the roots of $P(x)$.
Step 2: The roots of $P(x-2)$ are then $a + 2, b + 2, c + 2,$ so the product of the roots of $P(x-2)$ is the area of the desired rectangular prism.
Step 3: $P(x-2)$ has leading coefficient $10$ and constant term $P(0-2) = P(-2) = 10(-2)^3 - 39(-2)^2 + 29(-2) - 6 = -300$.
Step 4: Thus, by Vieta's Formulas, the product of the roots of $P(x-2)$ is $\frac{-(-300)}{10} = 30$.
Final Answer: \boxed{30} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: Cyclic quadrilateral $ABCD$ has lengths $BC = CD = 3$ and $DA = 5$ with $\angle CDA = 120^\circ$. What is the length of the shorter diagonal of $ABCD$?
Response:

Output: \frac{39}{7}


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. $\frac{p}{q}=\cos(\angle CMD)$ is irreducible fraction, what is the value of $p+q$?
Response:
Step 1: Without loss of generality, let the edge-length of $ABCD$ be $2.$ It follows that $MC=MD=\sqrt3.$
Step 2: Let $O$ be the center of $\triangle ABD,$ so $\overline{CO}\perp\overline{MOD}.$ Note that $MO=\frac13 MD=\frac{\sqrt{3}}{3}.$
Step 3: In right $\triangle CMO,$ we have\[\cos(\angle CMD)=\frac{MO}{MC}=\frac13.\]
Step 4: So the answer is $1+3=4.$
Final Answer: \boxed{4} ####

Question: How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser numbers?
Response:
Step 1: Clearly, the integers from $8$ through $14$ must be in different pairs, so are the integers from $1$ through $7.$ Note that $7$ must pair with $14.$
Step 2: We pair the numbers $1,2,3,4,5,6$ with the numbers $8,9,10,11,12,13$ systematically.
Step 3: $6$ can pair with either $12$ or $13.$
Step 4: $5$ can pair with any of the three remaining numbers from $10,11,12,13.$
Step 5: $1,2,3,4$ can pair with the other four remaining numbers from $8,9,10,11,12,13$ without restrictions.
Step 6: Together, the answer is $2\cdot3\cdot4!=144.$
Final Answer: \boxed{144} ####

Question: The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box?
Response:
Step 1: Let $P(x) = 10x^3 - 39x^2 + 29x - 6$, and let $a, b, c$ be the roots of $P(x)$.
Step 2: The roots of $P(x-2)$ are then $a + 2, b + 2, c + 2,$ so the product of the roots of $P(x-2)$ is the area of the desired rectangular prism.
Step 3: $P(x-2)$ has leading coefficient $10$ and constant term $P(0-2) = P(-2) = 10(-2)^3 - 39(-2)^2 + 29(-2) - 6 = -300$.
Step 4: Thus, by Vieta's Formulas, the product of the roots of $P(x-2)$ is $\frac{-(-300)}{10} = 30$.
Final Answer: \boxed{30} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: Points $P$ and $Q$ are chosen uniformly and independently at random on sides $\overline{AB}$ and $\overline{AC}$, respectively, of equilateral triangle $\triangle ABC$. Which of the following intervals contains the probability that the area of $\triangle APQ$ is less than half the area of $\triangle ABC$?
Response:

Output: [\frac{3}{4}, \frac{7}{8}]


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. $\frac{p}{q}=\cos(\angle CMD)$ is irreducible fraction, what is the value of $p+q$?
Response:
Step 1: Without loss of generality, let the edge-length of $ABCD$ be $2.$ It follows that $MC=MD=\sqrt3.$
Step 2: Let $O$ be the center of $\triangle ABD,$ so $\overline{CO}\perp\overline{MOD}.$ Note that $MO=\frac13 MD=\frac{\sqrt{3}}{3}.$
Step 3: In right $\triangle CMO,$ we have\[\cos(\angle CMD)=\frac{MO}{MC}=\frac13.\]
Step 4: So the answer is $1+3=4.$
Final Answer: \boxed{4} ####

Question: How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser numbers?
Response:
Step 1: Clearly, the integers from $8$ through $14$ must be in different pairs, so are the integers from $1$ through $7.$ Note that $7$ must pair with $14.$
Step 2: We pair the numbers $1,2,3,4,5,6$ with the numbers $8,9,10,11,12,13$ systematically.
Step 3: $6$ can pair with either $12$ or $13.$
Step 4: $5$ can pair with any of the three remaining numbers from $10,11,12,13.$
Step 5: $1,2,3,4$ can pair with the other four remaining numbers from $8,9,10,11,12,13$ without restrictions.
Step 6: Together, the answer is $2\cdot3\cdot4!=144.$
Final Answer: \boxed{144} ####

Question: The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box?
Response:
Step 1: Let $P(x) = 10x^3 - 39x^2 + 29x - 6$, and let $a, b, c$ be the roots of $P(x)$.
Step 2: The roots of $P(x-2)$ are then $a + 2, b + 2, c + 2,$ so the product of the roots of $P(x-2)$ is the area of the desired rectangular prism.
Step 3: $P(x-2)$ has leading coefficient $10$ and constant term $P(0-2) = P(-2) = 10(-2)^3 - 39(-2)^2 + 29(-2) - 6 = -300$.
Step 4: Thus, by Vieta's Formulas, the product of the roots of $P(x-2)$ is $\frac{-(-300)}{10} = 30$.
Final Answer: \boxed{30} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: Suppose that $a_1 = 2$ and the sequence $(a_n)$ satisfies the recurrence relation $\frac{a_n - 1}{n - 1} = \frac{a_{n-1} + 1}{(n - 1) + 1}$ for all $n \geq 2$. What is the greatest integer less than or equal to $\sum_{n=1}^{100} a_n^2$?
Response:

Output: 338551


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. $\frac{p}{q}=\cos(\angle CMD)$ is irreducible fraction, what is the value of $p+q$?
Response:
Step 1: Without loss of generality, let the edge-length of $ABCD$ be $2.$ It follows that $MC=MD=\sqrt3.$
Step 2: Let $O$ be the center of $\triangle ABD,$ so $\overline{CO}\perp\overline{MOD}.$ Note that $MO=\frac13 MD=\frac{\sqrt{3}}{3}.$
Step 3: In right $\triangle CMO,$ we have\[\cos(\angle CMD)=\frac{MO}{MC}=\frac13.\]
Step 4: So the answer is $1+3=4.$
Final Answer: \boxed{4} ####

Question: How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser numbers?
Response:
Step 1: Clearly, the integers from $8$ through $14$ must be in different pairs, so are the integers from $1$ through $7.$ Note that $7$ must pair with $14.$
Step 2: We pair the numbers $1,2,3,4,5,6$ with the numbers $8,9,10,11,12,13$ systematically.
Step 3: $6$ can pair with either $12$ or $13.$
Step 4: $5$ can pair with any of the three remaining numbers from $10,11,12,13.$
Step 5: $1,2,3,4$ can pair with the other four remaining numbers from $8,9,10,11,12,13$ without restrictions.
Step 6: Together, the answer is $2\cdot3\cdot4!=144.$
Final Answer: \boxed{144} ####

Question: The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box?
Response:
Step 1: Let $P(x) = 10x^3 - 39x^2 + 29x - 6$, and let $a, b, c$ be the roots of $P(x)$.
Step 2: The roots of $P(x-2)$ are then $a + 2, b + 2, c + 2,$ so the product of the roots of $P(x-2)$ is the area of the desired rectangular prism.
Step 3: $P(x-2)$ has leading coefficient $10$ and constant term $P(0-2) = P(-2) = 10(-2)^3 - 39(-2)^2 + 29(-2) - 6 = -300$.
Step 4: Thus, by Vieta's Formulas, the product of the roots of $P(x-2)$ is $\frac{-(-300)}{10} = 30$.
Final Answer: \boxed{30} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: What is the value of $\tan^2 \frac{\pi}{16}\cdot\tan^2 \frac{3\pi}{16} + \tan^2 \frac{\pi}{16}\cdot\tan^2 \frac{5\pi}{16} + \tan^2 \frac{3\pi}{16}\cdot\tan^2 \frac{7\pi}{16} + \tan^2 \frac{5\pi}{16}\cdot\tan^2 \frac{7\pi}{16}$?
Response:

Output: 68


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. $\frac{p}{q}=\cos(\angle CMD)$ is irreducible fraction, what is the value of $p+q$?
Response:
Step 1: Without loss of generality, let the edge-length of $ABCD$ be $2.$ It follows that $MC=MD=\sqrt3.$
Step 2: Let $O$ be the center of $\triangle ABD,$ so $\overline{CO}\perp\overline{MOD}.$ Note that $MO=\frac13 MD=\frac{\sqrt{3}}{3}.$
Step 3: In right $\triangle CMO,$ we have\[\cos(\angle CMD)=\frac{MO}{MC}=\frac13.\]
Step 4: So the answer is $1+3=4.$
Final Answer: \boxed{4} ####

Question: How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser numbers?
Response:
Step 1: Clearly, the integers from $8$ through $14$ must be in different pairs, so are the integers from $1$ through $7.$ Note that $7$ must pair with $14.$
Step 2: We pair the numbers $1,2,3,4,5,6$ with the numbers $8,9,10,11,12,13$ systematically.
Step 3: $6$ can pair with either $12$ or $13.$
Step 4: $5$ can pair with any of the three remaining numbers from $10,11,12,13.$
Step 5: $1,2,3,4$ can pair with the other four remaining numbers from $8,9,10,11,12,13$ without restrictions.
Step 6: Together, the answer is $2\cdot3\cdot4!=144.$
Final Answer: \boxed{144} ####

Question: The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box?
Response:
Step 1: Let $P(x) = 10x^3 - 39x^2 + 29x - 6$, and let $a, b, c$ be the roots of $P(x)$.
Step 2: The roots of $P(x-2)$ are then $a + 2, b + 2, c + 2,$ so the product of the roots of $P(x-2)$ is the area of the desired rectangular prism.
Step 3: $P(x-2)$ has leading coefficient $10$ and constant term $P(0-2) = P(-2) = 10(-2)^3 - 39(-2)^2 + 29(-2) - 6 = -300$.
Step 4: Thus, by Vieta's Formulas, the product of the roots of $P(x-2)$ is $\frac{-(-300)}{10} = 30$.
Final Answer: \boxed{30} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: A disphenoid is a tetrahedron whose triangular faces are congruent to one another. What is the least total surface area of a disphenoid whose faces are scalene triangles with integer side lengths?
Response:

Output: 15\sqrt{7}


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. $\frac{p}{q}=\cos(\angle CMD)$ is irreducible fraction, what is the value of $p+q$?
Response:
Step 1: Without loss of generality, let the edge-length of $ABCD$ be $2.$ It follows that $MC=MD=\sqrt3.$
Step 2: Let $O$ be the center of $\triangle ABD,$ so $\overline{CO}\perp\overline{MOD}.$ Note that $MO=\frac13 MD=\frac{\sqrt{3}}{3}.$
Step 3: In right $\triangle CMO,$ we have\[\cos(\angle CMD)=\frac{MO}{MC}=\frac13.\]
Step 4: So the answer is $1+3=4.$
Final Answer: \boxed{4} ####

Question: How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser numbers?
Response:
Step 1: Clearly, the integers from $8$ through $14$ must be in different pairs, so are the integers from $1$ through $7.$ Note that $7$ must pair with $14.$
Step 2: We pair the numbers $1,2,3,4,5,6$ with the numbers $8,9,10,11,12,13$ systematically.
Step 3: $6$ can pair with either $12$ or $13.$
Step 4: $5$ can pair with any of the three remaining numbers from $10,11,12,13.$
Step 5: $1,2,3,4$ can pair with the other four remaining numbers from $8,9,10,11,12,13$ without restrictions.
Step 6: Together, the answer is $2\cdot3\cdot4!=144.$
Final Answer: \boxed{144} ####

Question: The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box?
Response:
Step 1: Let $P(x) = 10x^3 - 39x^2 + 29x - 6$, and let $a, b, c$ be the roots of $P(x)$.
Step 2: The roots of $P(x-2)$ are then $a + 2, b + 2, c + 2,$ so the product of the roots of $P(x-2)$ is the area of the desired rectangular prism.
Step 3: $P(x-2)$ has leading coefficient $10$ and constant term $P(0-2) = P(-2) = 10(-2)^3 - 39(-2)^2 + 29(-2) - 6 = -300$.
Step 4: Thus, by Vieta's Formulas, the product of the roots of $P(x-2)$ is $\frac{-(-300)}{10} = 30$.
Final Answer: \boxed{30} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: A graph is symmetric about a line if the graph remains unchanged after reflection in that line. For how many quadruples of integers $(a, b, c, d)$, where $|a|, |b|, |c|, |d| \leq 5$ and $c$ and $d$ are not both $0$, is the graph of $y = \frac{ax + b}{cx + d}$ symmetric about the line $y = x$?
Response:

Output: 1292


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. $\frac{p}{q}=\cos(\angle CMD)$ is irreducible fraction, what is the value of $p+q$?
Response:
Step 1: Without loss of generality, let the edge-length of $ABCD$ be $2.$ It follows that $MC=MD=\sqrt3.$
Step 2: Let $O$ be the center of $\triangle ABD,$ so $\overline{CO}\perp\overline{MOD}.$ Note that $MO=\frac13 MD=\frac{\sqrt{3}}{3}.$
Step 3: In right $\triangle CMO,$ we have\[\cos(\angle CMD)=\frac{MO}{MC}=\frac13.\]
Step 4: So the answer is $1+3=4.$
Final Answer: \boxed{4} ####

Question: How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser numbers?
Response:
Step 1: Clearly, the integers from $8$ through $14$ must be in different pairs, so are the integers from $1$ through $7.$ Note that $7$ must pair with $14.$
Step 2: We pair the numbers $1,2,3,4,5,6$ with the numbers $8,9,10,11,12,13$ systematically.
Step 3: $6$ can pair with either $12$ or $13.$
Step 4: $5$ can pair with any of the three remaining numbers from $10,11,12,13.$
Step 5: $1,2,3,4$ can pair with the other four remaining numbers from $8,9,10,11,12,13$ without restrictions.
Step 6: Together, the answer is $2\cdot3\cdot4!=144.$
Final Answer: \boxed{144} ####

Question: The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box?
Response:
Step 1: Let $P(x) = 10x^3 - 39x^2 + 29x - 6$, and let $a, b, c$ be the roots of $P(x)$.
Step 2: The roots of $P(x-2)$ are then $a + 2, b + 2, c + 2,$ so the product of the roots of $P(x-2)$ is the area of the desired rectangular prism.
Step 3: $P(x-2)$ has leading coefficient $10$ and constant term $P(0-2) = P(-2) = 10(-2)^3 - 39(-2)^2 + 29(-2) - 6 = -300$.
Step 4: Thus, by Vieta's Formulas, the product of the roots of $P(x-2)$ is $\frac{-(-300)}{10} = 30$.
Final Answer: \boxed{30} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: In a long line of people arranged left to right, the 1013th person from the left is also the 1010th person from the right. How many people are in the line?
Response:

Output: 2022


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. $\frac{p}{q}=\cos(\angle CMD)$ is irreducible fraction, what is the value of $p+q$?
Response:
Step 1: Without loss of generality, let the edge-length of $ABCD$ be $2.$ It follows that $MC=MD=\sqrt3.$
Step 2: Let $O$ be the center of $\triangle ABD,$ so $\overline{CO}\perp\overline{MOD}.$ Note that $MO=\frac13 MD=\frac{\sqrt{3}}{3}.$
Step 3: In right $\triangle CMO,$ we have\[\cos(\angle CMD)=\frac{MO}{MC}=\frac13.\]
Step 4: So the answer is $1+3=4.$
Final Answer: \boxed{4} ####

Question: How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser numbers?
Response:
Step 1: Clearly, the integers from $8$ through $14$ must be in different pairs, so are the integers from $1$ through $7.$ Note that $7$ must pair with $14.$
Step 2: We pair the numbers $1,2,3,4,5,6$ with the numbers $8,9,10,11,12,13$ systematically.
Step 3: $6$ can pair with either $12$ or $13.$
Step 4: $5$ can pair with any of the three remaining numbers from $10,11,12,13.$
Step 5: $1,2,3,4$ can pair with the other four remaining numbers from $8,9,10,11,12,13$ without restrictions.
Step 6: Together, the answer is $2\cdot3\cdot4!=144.$
Final Answer: \boxed{144} ####

Question: The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box?
Response:
Step 1: Let $P(x) = 10x^3 - 39x^2 + 29x - 6$, and let $a, b, c$ be the roots of $P(x)$.
Step 2: The roots of $P(x-2)$ are then $a + 2, b + 2, c + 2,$ so the product of the roots of $P(x-2)$ is the area of the desired rectangular prism.
Step 3: $P(x-2)$ has leading coefficient $10$ and constant term $P(0-2) = P(-2) = 10(-2)^3 - 39(-2)^2 + 29(-2) - 6 = -300$.
Step 4: Thus, by Vieta's Formulas, the product of the roots of $P(x-2)$ is $\frac{-(-300)}{10} = 30$.
Final Answer: \boxed{30} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: What is $10! - 7! \cdot 6!$?
Response:

Output: 0


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. $\frac{p}{q}=\cos(\angle CMD)$ is irreducible fraction, what is the value of $p+q$?
Response:
Step 1: Without loss of generality, let the edge-length of $ABCD$ be $2.$ It follows that $MC=MD=\sqrt3.$
Step 2: Let $O$ be the center of $\triangle ABD,$ so $\overline{CO}\perp\overline{MOD}.$ Note that $MO=\frac13 MD=\frac{\sqrt{3}}{3}.$
Step 3: In right $\triangle CMO,$ we have\[\cos(\angle CMD)=\frac{MO}{MC}=\frac13.\]
Step 4: So the answer is $1+3=4.$
Final Answer: \boxed{4} ####

Question: How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser numbers?
Response:
Step 1: Clearly, the integers from $8$ through $14$ must be in different pairs, so are the integers from $1$ through $7.$ Note that $7$ must pair with $14.$
Step 2: We pair the numbers $1,2,3,4,5,6$ with the numbers $8,9,10,11,12,13$ systematically.
Step 3: $6$ can pair with either $12$ or $13.$
Step 4: $5$ can pair with any of the three remaining numbers from $10,11,12,13.$
Step 5: $1,2,3,4$ can pair with the other four remaining numbers from $8,9,10,11,12,13$ without restrictions.
Step 6: Together, the answer is $2\cdot3\cdot4!=144.$
Final Answer: \boxed{144} ####

Question: The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box?
Response:
Step 1: Let $P(x) = 10x^3 - 39x^2 + 29x - 6$, and let $a, b, c$ be the roots of $P(x)$.
Step 2: The roots of $P(x-2)$ are then $a + 2, b + 2, c + 2,$ so the product of the roots of $P(x-2)$ is the area of the desired rectangular prism.
Step 3: $P(x-2)$ has leading coefficient $10$ and constant term $P(0-2) = P(-2) = 10(-2)^3 - 39(-2)^2 + 29(-2) - 6 = -300$.
Step 4: Thus, by Vieta's Formulas, the product of the roots of $P(x-2)$ is $\frac{-(-300)}{10} = 30$.
Final Answer: \boxed{30} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: For how many integer values of $x$ is $|2x| \leq 7\pi$?
Response:

Output: 21


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. $\frac{p}{q}=\cos(\angle CMD)$ is irreducible fraction, what is the value of $p+q$?
Response:
Step 1: Without loss of generality, let the edge-length of $ABCD$ be $2.$ It follows that $MC=MD=\sqrt3.$
Step 2: Let $O$ be the center of $\triangle ABD,$ so $\overline{CO}\perp\overline{MOD}.$ Note that $MO=\frac13 MD=\frac{\sqrt{3}}{3}.$
Step 3: In right $\triangle CMO,$ we have\[\cos(\angle CMD)=\frac{MO}{MC}=\frac13.\]
Step 4: So the answer is $1+3=4.$
Final Answer: \boxed{4} ####

Question: How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser numbers?
Response:
Step 1: Clearly, the integers from $8$ through $14$ must be in different pairs, so are the integers from $1$ through $7.$ Note that $7$ must pair with $14.$
Step 2: We pair the numbers $1,2,3,4,5,6$ with the numbers $8,9,10,11,12,13$ systematically.
Step 3: $6$ can pair with either $12$ or $13.$
Step 4: $5$ can pair with any of the three remaining numbers from $10,11,12,13.$
Step 5: $1,2,3,4$ can pair with the other four remaining numbers from $8,9,10,11,12,13$ without restrictions.
Step 6: Together, the answer is $2\cdot3\cdot4!=144.$
Final Answer: \boxed{144} ####

Question: The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box?
Response:
Step 1: Let $P(x) = 10x^3 - 39x^2 + 29x - 6$, and let $a, b, c$ be the roots of $P(x)$.
Step 2: The roots of $P(x-2)$ are then $a + 2, b + 2, c + 2,$ so the product of the roots of $P(x-2)$ is the area of the desired rectangular prism.
Step 3: $P(x-2)$ has leading coefficient $10$ and constant term $P(0-2) = P(-2) = 10(-2)^3 - 39(-2)^2 + 29(-2) - 6 = -300$.
Step 4: Thus, by Vieta's Formulas, the product of the roots of $P(x-2)$ is $\frac{-(-300)}{10} = 30$.
Final Answer: \boxed{30} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: Balls numbered $1, 2, 3, \ldots$ are deposited in $5$ bins, labeled $A, B, C, D$, and $E$, using the following procedure. Ball $1$ is deposited in bin $A$, and balls $2$ and $3$ are deposited in $B$. The next three balls are deposited in bin $C$, the next $4$ in bin $D$, and so on, cycling back to bin $A$ after balls are deposited in bin $E$. (For example, $22, 23, \ldots, 28$ are deposited in bin $B$ at step 7 of this process.) In which bin is ball $2024$ deposited?
Response:

Output: D


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. $\frac{p}{q}=\cos(\angle CMD)$ is irreducible fraction, what is the value of $p+q$?
Response:
Step 1: Without loss of generality, let the edge-length of $ABCD$ be $2.$ It follows that $MC=MD=\sqrt3.$
Step 2: Let $O$ be the center of $\triangle ABD,$ so $\overline{CO}\perp\overline{MOD}.$ Note that $MO=\frac13 MD=\frac{\sqrt{3}}{3}.$
Step 3: In right $\triangle CMO,$ we have\[\cos(\angle CMD)=\frac{MO}{MC}=\frac13.\]
Step 4: So the answer is $1+3=4.$
Final Answer: \boxed{4} ####

Question: How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser numbers?
Response:
Step 1: Clearly, the integers from $8$ through $14$ must be in different pairs, so are the integers from $1$ through $7.$ Note that $7$ must pair with $14.$
Step 2: We pair the numbers $1,2,3,4,5,6$ with the numbers $8,9,10,11,12,13$ systematically.
Step 3: $6$ can pair with either $12$ or $13.$
Step 4: $5$ can pair with any of the three remaining numbers from $10,11,12,13.$
Step 5: $1,2,3,4$ can pair with the other four remaining numbers from $8,9,10,11,12,13$ without restrictions.
Step 6: Together, the answer is $2\cdot3\cdot4!=144.$
Final Answer: \boxed{144} ####

Question: The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box?
Response:
Step 1: Let $P(x) = 10x^3 - 39x^2 + 29x - 6$, and let $a, b, c$ be the roots of $P(x)$.
Step 2: The roots of $P(x-2)$ are then $a + 2, b + 2, c + 2,$ so the product of the roots of $P(x-2)$ is the area of the desired rectangular prism.
Step 3: $P(x-2)$ has leading coefficient $10$ and constant term $P(0-2) = P(-2) = 10(-2)^3 - 39(-2)^2 + 29(-2) - 6 = -300$.
Step 4: Thus, by Vieta's Formulas, the product of the roots of $P(x-2)$ is $\frac{-(-300)}{10} = 30$.
Final Answer: \boxed{30} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: In the following expression, Melanie changed some of the plus signs to minus signs: $1 + 3 + 5 + 7 + \cdots + 97 + 99$. When the new expression was evaluated, it was negative. What is the least number of plus signs that Melanie could have changed to minus signs?
Response:

Output: 15


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. $\frac{p}{q}=\cos(\angle CMD)$ is irreducible fraction, what is the value of $p+q$?
Response:
Step 1: Without loss of generality, let the edge-length of $ABCD$ be $2.$ It follows that $MC=MD=\sqrt3.$
Step 2: Let $O$ be the center of $\triangle ABD,$ so $\overline{CO}\perp\overline{MOD}.$ Note that $MO=\frac13 MD=\frac{\sqrt{3}}{3}.$
Step 3: In right $\triangle CMO,$ we have\[\cos(\angle CMD)=\frac{MO}{MC}=\frac13.\]
Step 4: So the answer is $1+3=4.$
Final Answer: \boxed{4} ####

Question: How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser numbers?
Response:
Step 1: Clearly, the integers from $8$ through $14$ must be in different pairs, so are the integers from $1$ through $7.$ Note that $7$ must pair with $14.$
Step 2: We pair the numbers $1,2,3,4,5,6$ with the numbers $8,9,10,11,12,13$ systematically.
Step 3: $6$ can pair with either $12$ or $13.$
Step 4: $5$ can pair with any of the three remaining numbers from $10,11,12,13.$
Step 5: $1,2,3,4$ can pair with the other four remaining numbers from $8,9,10,11,12,13$ without restrictions.
Step 6: Together, the answer is $2\cdot3\cdot4!=144.$
Final Answer: \boxed{144} ####

Question: The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box?
Response:
Step 1: Let $P(x) = 10x^3 - 39x^2 + 29x - 6$, and let $a, b, c$ be the roots of $P(x)$.
Step 2: The roots of $P(x-2)$ are then $a + 2, b + 2, c + 2,$ so the product of the roots of $P(x-2)$ is the area of the desired rectangular prism.
Step 3: $P(x-2)$ has leading coefficient $10$ and constant term $P(0-2) = P(-2) = 10(-2)^3 - 39(-2)^2 + 29(-2) - 6 = -300$.
Step 4: Thus, by Vieta's Formulas, the product of the roots of $P(x-2)$ is $\frac{-(-300)}{10} = 30$.
Final Answer: \boxed{30} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: The national debt of the United States is on track to reach $5 \cdot 10^{13}$ dollars by $2033$. How many digits does this number of dollars have when written as a numeral in base $5$? (The approximation of $\log_{10} 5$ as $0.7$ is sufficient for this problem.)
Response:

Output: 20


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. $\frac{p}{q}=\cos(\angle CMD)$ is irreducible fraction, what is the value of $p+q$?
Response:
Step 1: Without loss of generality, let the edge-length of $ABCD$ be $2.$ It follows that $MC=MD=\sqrt3.$
Step 2: Let $O$ be the center of $\triangle ABD,$ so $\overline{CO}\perp\overline{MOD}.$ Note that $MO=\frac13 MD=\frac{\sqrt{3}}{3}.$
Step 3: In right $\triangle CMO,$ we have\[\cos(\angle CMD)=\frac{MO}{MC}=\frac13.\]
Step 4: So the answer is $1+3=4.$
Final Answer: \boxed{4} ####

Question: How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser numbers?
Response:
Step 1: Clearly, the integers from $8$ through $14$ must be in different pairs, so are the integers from $1$ through $7.$ Note that $7$ must pair with $14.$
Step 2: We pair the numbers $1,2,3,4,5,6$ with the numbers $8,9,10,11,12,13$ systematically.
Step 3: $6$ can pair with either $12$ or $13.$
Step 4: $5$ can pair with any of the three remaining numbers from $10,11,12,13.$
Step 5: $1,2,3,4$ can pair with the other four remaining numbers from $8,9,10,11,12,13$ without restrictions.
Step 6: Together, the answer is $2\cdot3\cdot4!=144.$
Final Answer: \boxed{144} ####

Question: The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box?
Response:
Step 1: Let $P(x) = 10x^3 - 39x^2 + 29x - 6$, and let $a, b, c$ be the roots of $P(x)$.
Step 2: The roots of $P(x-2)$ are then $a + 2, b + 2, c + 2,$ so the product of the roots of $P(x-2)$ is the area of the desired rectangular prism.
Step 3: $P(x-2)$ has leading coefficient $10$ and constant term $P(0-2) = P(-2) = 10(-2)^3 - 39(-2)^2 + 29(-2) - 6 = -300$.
Step 4: Thus, by Vieta's Formulas, the product of the roots of $P(x-2)$ is $\frac{-(-300)}{10} = 30$.
Final Answer: \boxed{30} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: What value of $x$ satisfies $\frac{\log_2 x \cdot \log_3 x}{\log_2 x + \log_3 x} = 2$?
Response:

Output: 36


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. $\frac{p}{q}=\cos(\angle CMD)$ is irreducible fraction, what is the value of $p+q$?
Response:
Step 1: Without loss of generality, let the edge-length of $ABCD$ be $2.$ It follows that $MC=MD=\sqrt3.$
Step 2: Let $O$ be the center of $\triangle ABD,$ so $\overline{CO}\perp\overline{MOD}.$ Note that $MO=\frac13 MD=\frac{\sqrt{3}}{3}.$
Step 3: In right $\triangle CMO,$ we have\[\cos(\angle CMD)=\frac{MO}{MC}=\frac13.\]
Step 4: So the answer is $1+3=4.$
Final Answer: \boxed{4} ####

Question: How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser numbers?
Response:
Step 1: Clearly, the integers from $8$ through $14$ must be in different pairs, so are the integers from $1$ through $7.$ Note that $7$ must pair with $14.$
Step 2: We pair the numbers $1,2,3,4,5,6$ with the numbers $8,9,10,11,12,13$ systematically.
Step 3: $6$ can pair with either $12$ or $13.$
Step 4: $5$ can pair with any of the three remaining numbers from $10,11,12,13.$
Step 5: $1,2,3,4$ can pair with the other four remaining numbers from $8,9,10,11,12,13$ without restrictions.
Step 6: Together, the answer is $2\cdot3\cdot4!=144.$
Final Answer: \boxed{144} ####

Question: The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box?
Response:
Step 1: Let $P(x) = 10x^3 - 39x^2 + 29x - 6$, and let $a, b, c$ be the roots of $P(x)$.
Step 2: The roots of $P(x-2)$ are then $a + 2, b + 2, c + 2,$ so the product of the roots of $P(x-2)$ is the area of the desired rectangular prism.
Step 3: $P(x-2)$ has leading coefficient $10$ and constant term $P(0-2) = P(-2) = 10(-2)^3 - 39(-2)^2 + 29(-2) - 6 = -300$.
Step 4: Thus, by Vieta's Formulas, the product of the roots of $P(x-2)$ is $\frac{-(-300)}{10} = 30$.
Final Answer: \boxed{30} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: A dartboard is the region $B$ in the coordinate plane consisting of points $(x, y)$ such that $|x| + |y| \leq 8$. A target $T$ is the region where $(x^2 + y^2 - 25)^2 \leq 49$. A dart is thrown and lands at a random point in $B$. The probability that the dart lands in $T$ can be expressed as $\frac{m}{n} \cdot \pi$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$?
Response:

Output: 71


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. $\frac{p}{q}=\cos(\angle CMD)$ is irreducible fraction, what is the value of $p+q$?
Response:
Step 1: Without loss of generality, let the edge-length of $ABCD$ be $2.$ It follows that $MC=MD=\sqrt3.$
Step 2: Let $O$ be the center of $\triangle ABD,$ so $\overline{CO}\perp\overline{MOD}.$ Note that $MO=\frac13 MD=\frac{\sqrt{3}}{3}.$
Step 3: In right $\triangle CMO,$ we have\[\cos(\angle CMD)=\frac{MO}{MC}=\frac13.\]
Step 4: So the answer is $1+3=4.$
Final Answer: \boxed{4} ####

Question: How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser numbers?
Response:
Step 1: Clearly, the integers from $8$ through $14$ must be in different pairs, so are the integers from $1$ through $7.$ Note that $7$ must pair with $14.$
Step 2: We pair the numbers $1,2,3,4,5,6$ with the numbers $8,9,10,11,12,13$ systematically.
Step 3: $6$ can pair with either $12$ or $13.$
Step 4: $5$ can pair with any of the three remaining numbers from $10,11,12,13.$
Step 5: $1,2,3,4$ can pair with the other four remaining numbers from $8,9,10,11,12,13$ without restrictions.
Step 6: Together, the answer is $2\cdot3\cdot4!=144.$
Final Answer: \boxed{144} ####

Question: The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box?
Response:
Step 1: Let $P(x) = 10x^3 - 39x^2 + 29x - 6$, and let $a, b, c$ be the roots of $P(x)$.
Step 2: The roots of $P(x-2)$ are then $a + 2, b + 2, c + 2,$ so the product of the roots of $P(x-2)$ is the area of the desired rectangular prism.
Step 3: $P(x-2)$ has leading coefficient $10$ and constant term $P(0-2) = P(-2) = 10(-2)^3 - 39(-2)^2 + 29(-2) - 6 = -300$.
Step 4: Thus, by Vieta's Formulas, the product of the roots of $P(x-2)$ is $\frac{-(-300)}{10} = 30$.
Final Answer: \boxed{30} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: A list of 9 real numbers consists of $1, 2, 2, 3, 2, 5, 2, 6, 2$ and $7$, as well as $x, y, z$ with $x \leq y \leq z$. The range of the list is $7$, and the mean and median are both positive integers. How many ordered triples $(x, y, z)$ are possible?
Response:

Output: 3


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. $\frac{p}{q}=\cos(\angle CMD)$ is irreducible fraction, what is the value of $p+q$?
Response:
Step 1: Without loss of generality, let the edge-length of $ABCD$ be $2.$ It follows that $MC=MD=\sqrt3.$
Step 2: Let $O$ be the center of $\triangle ABD,$ so $\overline{CO}\perp\overline{MOD}.$ Note that $MO=\frac13 MD=\frac{\sqrt{3}}{3}.$
Step 3: In right $\triangle CMO,$ we have\[\cos(\angle CMD)=\frac{MO}{MC}=\frac13.\]
Step 4: So the answer is $1+3=4.$
Final Answer: \boxed{4} ####

Question: How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser numbers?
Response:
Step 1: Clearly, the integers from $8$ through $14$ must be in different pairs, so are the integers from $1$ through $7.$ Note that $7$ must pair with $14.$
Step 2: We pair the numbers $1,2,3,4,5,6$ with the numbers $8,9,10,11,12,13$ systematically.
Step 3: $6$ can pair with either $12$ or $13.$
Step 4: $5$ can pair with any of the three remaining numbers from $10,11,12,13.$
Step 5: $1,2,3,4$ can pair with the other four remaining numbers from $8,9,10,11,12,13$ without restrictions.
Step 6: Together, the answer is $2\cdot3\cdot4!=144.$
Final Answer: \boxed{144} ####

Question: The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box?
Response:
Step 1: Let $P(x) = 10x^3 - 39x^2 + 29x - 6$, and let $a, b, c$ be the roots of $P(x)$.
Step 2: The roots of $P(x-2)$ are then $a + 2, b + 2, c + 2,$ so the product of the roots of $P(x-2)$ is the area of the desired rectangular prism.
Step 3: $P(x-2)$ has leading coefficient $10$ and constant term $P(0-2) = P(-2) = 10(-2)^3 - 39(-2)^2 + 29(-2) - 6 = -300$.
Step 4: Thus, by Vieta's Formulas, the product of the roots of $P(x-2)$ is $\frac{-(-300)}{10} = 30$.
Final Answer: \boxed{30} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: Let $x_n = \sin^2 (n^5)$. What is the mean of $x_1, x_2, x_3, \cdots, x_{90}$?
Response:

Output: \frac{91}{180}


Input:
Answer the following question. Break down your reasoning process into small steps. Each step should represent a single, minimal reasoning action, and each step must logically follow the previous one.
Use the following format for each step:
Step i: [Your reasoning process in one cohesive response]
After completing all the steps, conclude with:
Final Answer: \boxed{[Your final answer here without the unit or any additional text]} ####
Stop generation immediately after outputing the Final Answer.

The following are 3 examples:

Question: Let $M$ be the midpoint of $\overline{AB}$ in regular tetrahedron $ABCD$. $\frac{p}{q}=\cos(\angle CMD)$ is irreducible fraction, what is the value of $p+q$?
Response:
Step 1: Without loss of generality, let the edge-length of $ABCD$ be $2.$ It follows that $MC=MD=\sqrt3.$
Step 2: Let $O$ be the center of $\triangle ABD,$ so $\overline{CO}\perp\overline{MOD}.$ Note that $MO=\frac13 MD=\frac{\sqrt{3}}{3}.$
Step 3: In right $\triangle CMO,$ we have\[\cos(\angle CMD)=\frac{MO}{MC}=\frac13.\]
Step 4: So the answer is $1+3=4.$
Final Answer: \boxed{4} ####

Question: How many ways are there to split the integers $1$ through $14$ into $7$ pairs such that in each pair, the greater number is at least $2$ times the lesser numbers?
Response:
Step 1: Clearly, the integers from $8$ through $14$ must be in different pairs, so are the integers from $1$ through $7.$ Note that $7$ must pair with $14.$
Step 2: We pair the numbers $1,2,3,4,5,6$ with the numbers $8,9,10,11,12,13$ systematically.
Step 3: $6$ can pair with either $12$ or $13.$
Step 4: $5$ can pair with any of the three remaining numbers from $10,11,12,13.$
Step 5: $1,2,3,4$ can pair with the other four remaining numbers from $8,9,10,11,12,13$ without restrictions.
Step 6: Together, the answer is $2\cdot3\cdot4!=144.$
Final Answer: \boxed{144} ####

Question: The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by $2$ units. What is the volume of the new box?
Response:
Step 1: Let $P(x) = 10x^3 - 39x^2 + 29x - 6$, and let $a, b, c$ be the roots of $P(x)$.
Step 2: The roots of $P(x-2)$ are then $a + 2, b + 2, c + 2,$ so the product of the roots of $P(x-2)$ is the area of the desired rectangular prism.
Step 3: $P(x-2)$ has leading coefficient $10$ and constant term $P(0-2) = P(-2) = 10(-2)^3 - 39(-2)^2 + 29(-2) - 6 = -300$.
Step 4: Thus, by Vieta's Formulas, the product of the roots of $P(x-2)$ is $\frac{-(-300)}{10} = 30$.
Final Answer: \boxed{30} ####

Ensure that your response strictly follows the format to maintain clarity and consistency.
Question: Suppose $z$ is a complex number with positive imaginary part, with real part greater than $1$, and with $|z| = 2$. In the complex plane, the four points $0, z, z^2$, and $z^3$ are the vertices of a quadrilateral with area $15$. What is the imaginary part of $z$?
Response:

Output: \frac{3}{2}