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 real numbers $x, y, h$ and $k$ that satisfy the system of equations $x^2 + y^2 - 6x - 8y = h$ and $x^2 + y^2 - 10x + 4y = k$. What is the minimum possible value of $h + k$?
Response:

Output: -34


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 different remainders can result when the $100$th power of an integer is divided by $125$?
Response:

Output: 5


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 triangle in the coordinate plane has vertices $A(\log_2 1, \log_2 2), B(\log_2 3, \log_2 4)$, and $C(\log_2 7, \log_2 8)$. What is the area of $\triangle ABC$?
Response:

Output: \log_2 \frac{7}{\sqrt{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: A group of $16$ people will be partitioned into $4$ indistinguishable $4$-person committees. Each committee will have one chairperson and one secretary. The number of different ways to make these assignments can be written as $3^r M$, where $r$ and $M$ are positive integers and $M$ is not divisible by $3$. What is $r$?
Response:

Output: 5


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 $a$ and $b$ are randomly chosen without replacement from the set of integers with absolute value not exceeding $10$. What is the probability that the polynomial $x^3 + ax^2 + bx + 6$ has $3$ distinct integer roots?
Response:

Output: \frac{1}{105}


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 Fibonacci numbers are defined by $F_1 = 1, F_2 = 1$, and $F_n = F_{n-1} + F_{n-2}$ for $n \geq 3$. What is $\frac{F_2}{F_1} + \frac{F_4}{F_2} + \frac{F_6}{F_3} + \cdots + \frac{F_{20}}{F_{10}}$?
Response:

Output: 320


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 $A, B$, and $C$ are points in the plane with $AB = 40$ and $AC = 42$, and let $x$ be the length of the line segment from $A$ to the midpoint of $\overline{BC}$. Define a function $f$ by letting $f(x)$ be the area of $\triangle ABC$. Then the domain of $f$ is an open interval $(p, q)$, and the maximum value $r$ of $f(x)$ occurs at $x = 8$. What is $p + q + r + s$?
Response:

Output: 911


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 measures of the smallest angles of three different right triangles sum to $90^\circ$. All three triangles have side lengths that are primitive Pythagorean triples. Two of them are $3-4-5$ and $5-12-13$. What is the perimeter of the third triangle?
Response:

Output: 154


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 $\triangle ABC$ be a triangle with integer side lengths and the property that $\angle B = 2\angle A$. What is the least possible perimeter of such a triangle?
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: A right pyramid has regular octagon $ABCDEFGH$ with side length $1$ as its base and apex $V$. Segments $\overline{AV}$ and $\overline{DV}$ are perpendicular. What is the square of the height of the pyramid?
Response:

Output: \frac{1 + \sqrt{2}}{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: What is the number of ordered triples $(a, b, c)$ of positive integers, with $a \leq b \leq c \leq 9$, such that there exists a (non-degenerate) triangle $\triangle ABC$ with an integer inradius for which $a, b$, and $c$ are the lengths of the altitudes from $A$ to $\overline{BC}$, $B$ to $\overline{AC}$, and $C$ to $\overline{AB}$, respectively? (Recall that the inradius of a triangle is the radius of the largest possible circle that can be inscribed in the triangle.)
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: Pablo will decorate each of $6$ identical white balls with either a striped or a dotted pattern, using either red or blue paint. He will decide on the color and pattern for each ball by flipping a fair coin for each of the $12$ decisions he must make. After the paint dries, he will place the $6$ balls in an urn. Frida will randomly select one ball from the urn and note its color and pattern. The events "the ball Frida selects is red" and "the ball Frida selects is striped" may or may not be independent, depending on the outcome of Pablo's coin flips. The probability that these two events are independent can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m$? (Recall that two events $A$ and $B$ are independent if $P(A \text{ and } B) = P(A) \cdot P(B)$.)
Response:

Output: 247