Given a question, please decompose it into sub-questions. For each sub-question, please answer it in a complete sentence, ending with "The answer is". When the original question is answerable, please start the sub-question with "Now we can answer the question: ".

### Question: The function defined by
\[f(x) = \left\{
\begin{array}{cl}
x + k & \text{if $x < 4$}, \\
2x - 3 & \text{if $x \ge 4$}
\end{array}
\right.\]has an inverse, and the inverse is defined for all real numbers.  Enter all possible values of $k,$ separated by commas.

SubQuestion 1:

Response:
2x - 3 is the inverse of x + k if x < 4. What is the equation that sets up the inverse function?
Answer 1: To find the inverse of $x + k$ for $x < 4$, we set $y = x + k$ and solve for $x$ in terms of $y$. The inverse function is thus $x = y - k$. The answer is $x = y - k$.
SubQuestion 2: What is the equation that sets up the inverse function?
Answer 2: To find the inverse of $2x - 3$ for $x \ge 4$, we set $y = 2x - 3$ and solve for $x$ in terms of $y$. The inverse function is thus $x = \frac{y + 3}{2}$. The answer is $x = \frac{y + 3}{2}$.
SubQuestion 3: What is the range of $x$ for the inverse function to be defined?
Answer 3: The inverse function is defined for all real numbers, so the range of $x$ is all real numbers. The answer is all real numbers.
SubQuestion 4: What is the value of $k$ that makes the inverse function defined for $x < 4$?
Answer 4: To ensure the inverse is defined for $x < 4$, we set $y - k = x$ and solve for $k$. This gives $k = y - x = y - (y - k)$, which simplifies to $k = y$. Since $y$ is less than 4, the inverse is $x = y - k$, which means $k = y$. The answer is $k = y$.
SubQuestion 5: What is the value of $k$ that makes the inverse function defined for $x \ge 4$?
Answer 5: To ensure the inverse is defined for $x \ge 4$, we set $y = 2x - 3$ and solve for $x$. This gives $x = \frac{y + 3}{2}$, which is greater than or equal to 2 for $y \ge 4$. Thus, the inverse is $x = \frac{y + 3}{2}$, and since $y$ is greater than or equal to 4, the inverse is defined for all $y$. The answer is $k = \frac{y + 3}{2}$.
SubQuestion 6: What is the range of $k$ for the inverse function to be defined?
Answer 6: The range of $k$ is all real numbers since the inverse is defined for all $y$ for both cases. The answer is all real numbers.
SubQuestion 7: Now we can answer the question: What are all possible values of $k$?
Answer 7: The possible values of $k$ are all real numbers, which can be represented as a comma-separated list: any real number except for the value that makes the inverse undefined for $x < 4$, which is $y = 4$. The answer is any real number except 4. The answer is any real number except 4.


Given a question, please decompose it into sub-questions. For each sub-question, please answer it in a complete sentence, ending with "The answer is". When the original question is answerable, please start the sub-question with "Now we can answer the question: ".

### Question: Find constants $A,$ $B,$ $C,$ and $D$ so that
\[\frac{x^3 + 3x^2 - 12x + 36}{x^4 - 16} = \frac{A}{x - 2} + \frac{B}{x + 2} + \frac{Cx + D}{x^2 + 4}.\]Enter the ordered quadruple $(A,B,C,D).$

SubQuestion 1:

Response:
3x^2 - 12x + 36 can be factored into what?
Answer 1: 3x^2 - 12x + 36 can be factored into (3x - 6)(x - 6) which equals (3x - 6)(x - 2)(x + 3). The answer is (3x - 6)(x - 2)(x + 3).
SubQuestion 2: What is the common denominator of the right side of the equation?
Answer 2: The common denominator of the right side of the equation is (x - 2)(x + 2)(x + 4). The answer is (x - 2)(x + 2)(x + 4).
SubQuestion 3: Multiply both sides of the equation by the common denominator to clear the fractions.
Answer 3: Multiplying both sides by (x - 2)(x + 2)(x + 4) gives us:
\[(x^3 + 3x^2 - 12x + 36) = A(x + 2)(x + 4) + B(x - 2)(x + 3) + (Cx + D)(x - 2)(x + 2)(x + 4).\]The answer is (x^3 + 3x^2 - 12x + 36) = A(x + 2)(x + 4) + B(x - 2)(x + 3) + (Cx + D)(x - 2)(x + 2)(x + 4).
SubQuestion 4: Expand the right side of the equation.
Answer 4: Expanding the right side gives us:
\[(x^3 + 3x^2 - 12x + 36) = (A(x + 2)(x + 4) + B(x - 2)(x + 3) + (Cx + D)(x - 2)(x + 2)(x + 4))\]
\[= (A(x^2 + 2x + 4) + B(x^2 - x + 6) + (Cx + D)(x^2 + 4x + 8))\]
\[= (A + B + C)x^2 + (2A - B + 4D)x + (4A + 3B - 2C).\]The answer is (A + B + C)x^2 + (2A - B + 4D)x + (4A + 3B - 2C).
SubQuestion 5: Now we can answer the question: What are the values of A, B, C, and D?
Answer 5: By comparing coefficients, we get the following system of equations:
1. A + B + C = 1 (from x^2 terms)
2. 2A - B + 4D = 3 (from x terms)
3. 4A + 3B - 2C = -12 (from constant terms)
4. 4A + 3B - 2C = 36 (from constant terms)
Solving this system gives us:
A = 2
B = 1
C = -1
D = 2. The answer is (2, 1, -1, 2).


Given a question, please decompose it into sub-questions. For each sub-question, please answer it in a complete sentence, ending with "The answer is". When the original question is answerable, please start the sub-question with "Now we can answer the question: ".

### Question: A polynomial with integer coefficients is of the form
\[2x^3 + a_2 x^2 + a_1 x - 4 = 0.\]Find the number of possible rational roots of this polynomial.

SubQuestion 1:

Response:
2x^3 + a2x^2 + a1x - 4 = 0, what is the possible rational roots?
Answer 1: The possible rational roots of a polynomial with integer coefficients can be found using the Rational Root Theorem, which states that any rational root p/q (in lowest terms) of the polynomial must have p as a factor of the constant term and q as a factor of the leading coefficient. For the given polynomial, the constant term is -4 and the leading coefficient is 2. The factors of -4 are ±1, ±2, ±4. The answer is ±1, ±2, ±4.
SubQuestion 2: Now we can answer the question: How many possible rational roots does the polynomial 2x^3 + a2x^2 + a1x - 4 = 0 have?
Answer 2: The number of possible rational roots is equal to the number of factors of the constant term divided by the number of factors of the leading coefficient. In this case, there are 4 factors of -4 (±1, ±2, ±4) and 3 factors of 2 (±1, ±2). The number of possible rational roots is 4 / 3 = 2/1, which is 2. However, since the polynomial has integer coefficients, each rational root may have a corresponding conjugate root that is not listed in the Rational Root Theorem. Therefore, the total number of possible rational roots, including the conjugates, is 2 * 2 = 4. The answer is 4.


Given a question, please decompose it into sub-questions. For each sub-question, please answer it in a complete sentence, ending with "The answer is". When the original question is answerable, please start the sub-question with "Now we can answer the question: ".

### Question: Find the range of the function $f(x) = |x-4| + |x-20| + |x-50|,$ where $x$ is a real number.

SubQuestion 1:

Response:
4, 20, and 50 are the numbers that the expression inside the absolute values can be equal to. What is the minimum value of $x$ that makes any of these numbers equal to zero?
Answer 1: The minimum value of $x$ that makes $x-4=0$ is $x=4$. The minimum value of $x$ that makes $x-20=0$ is $x=20$. The minimum value of $x$ that makes $x-50=0$ is $x=50$. The answer is 4, 20, 50.
SubQuestion 2: What is the maximum value of $x$ that makes any of these numbers equal to zero?
Answer 2: The maximum value of $x$ that makes $x-4=0$ is $x=4$ (since $x$ would then be $4+4=8$ which is greater than $50$). The maximum value of $x$ that makes $x-20=0$ is $x=20$ (since $x$ would then be $20+20=40$ which is greater than $50$). The maximum value of $x$ that makes $x-50=0$ is $x=50$ (since $x$ would then be $50+50=100$ which is greater than $20$). The answer is 4, 20, 50.
SubQuestion 3: What is the minimum value of $x$ that makes any of the absolute values non-negative?
Answer 3: The minimum value of $x$ that makes $|x-4|$ non-negative is $x>4$. The minimum value of $x$ that makes $|x-20|$ non-negative is $x>20$. The minimum value of $x$ that makes $|x-50|$ non-negative is $x>50$. The answer is 4, 20, 50.
SubQuestion 4: What is the maximum value of $x$ that makes any of the absolute values non-positive?
Answer 4: The maximum value of $x$ that makes $|x-4|$ non-positive is $x<4$ (since $x$ would then be $4-4=0$ which is less than $20$). The maximum value of $x$ that makes $|x-20|$ non-positive is $x<20$ (since $x$ would then be $20-20=0$ which is less than $50$). The maximum value of $x$ that makes $|x-50|$ non-positive is $x<50$ (since $x$ would then be $50-50=0$ which is less than $20$). The answer is 4, 20, 50.
SubQuestion 5: Now we can answer the question: What is the range of the function $f(x)$?
Answer 5: The range of the function $f(x)$ is the set of all possible values that $f(x)$ can take. Since the absolute values are always non-negative and non-positive for all $x$, the minimum value of $f(x)$ occurs when all the absolute values are equal to zero, which is when $x=4$, $x=20$, and $x=50$. So the minimum value of $f(x)$ is $4+20+50=74$. The maximum value of $f(x)$ occurs when the terms inside the absolute values are as far apart as possible, which is when $x$ is as far from 4, 20, and 50 as possible, which is when $x$ is less than 4 and greater than 20 and 50. The maximum value of $f(x)$ is then $4 + 20 + 50 = 74$. Therefore, the range of $f(x)$ is $[74, 74]$. The answer is [74, 74].
SubQuestion 6: Now we can answer the question: What is the simplified form of the range of $f(x)$?
Answer 6: The simplified form of the range of $f(x)$ is $[74, 74]$, which is a single number. The answer is 74.


Given a question, please decompose it into sub-questions. For each sub-question, please answer it in a complete sentence, ending with "The answer is". When the original question is answerable, please start the sub-question with "Now we can answer the question: ".

### Question: Is the function $f(x) = 5^x - 5^{-x}$ even, odd, or neither?

Enter "odd", "even", or "neither".

SubQuestion 1:

Response:
5^x is an even function because it's a power of 5, and any power of 5 is even.
The answer is even.
SubQuestion 2: 5^-x is an odd function because it's the reciprocal of 5^x, and the reciprocal of an even number is odd.
The answer is odd.
SubQuestion 3: Now we can answer the question: Is the function $f(x) = 5^x - 5^{-x}$ even, odd, or neither?
The answer is odd.

###


Given a question, please decompose it into sub-questions. For each sub-question, please answer it in a complete sentence, ending with "The answer is". When the original question is answerable, please start the sub-question with "Now we can answer the question: ".

### Question: The sequence $\{a_n\}$ is defined by $$
a_0 = 1,a_1 = 1, \text{ and } a_n = a_{n - 1} + \frac {a_{n - 1}^2}{a_{n - 2}}\text{ for }n\ge2.
$$The sequence $\{b_n\}$ is defined by $$
b_0 = 1,b_1 = 3, \text{ and } b_n = b_{n - 1} + \frac {b_{n - 1}^2}{b_{n - 2}}\text{ for }n\ge2.$$Find $\frac {b_{32}}{a_{32}}$.

SubQuestion 1:

Response:
1. What is the value of $a_2$?
Answer 1: The value of $a_2$ is $a_1 + \frac {a_1^2}{a_0} = 1 + \frac {1}{1} = 2$. The answer is 2.
SubQuestion 2: 2. What is the value of $a_3$?
Answer 2: The value of $a_3$ is $a_2 + \frac {a_2^2}{a_1} = 2 + \frac {2^2}{1} = 2 + 4 = 6$. The answer is 6.
SubQuestion 3: 3. What is the value of $a_4$?
Answer 3: The value of $a_4$ is $a_3 + \frac {a_3^2}{a_2} = 6 + \frac {6^2}{2} = 6 + 18 = 24$. The answer is 24.
SubQuestion 4: 4. What is the value of $a_5$?
Answer 4: The value of $a_5$ is $a_4 + \frac {a_4^2}{a_3} = 24 + \frac {24^2}{6} = 24 + 144 = 168$. The answer is 168.
SubQuestion 5: 5. What is the value of $a_6$?
Answer 5: The value of $a_6$ is $a_5 + \frac {a_5^2}{a_4} = 168 + \frac {168^2}{24} = 168 + 1296 = 1464$. The answer is 1464.
SubQuestion 6: 6. What is the value of $a_7$?
Answer 6: The value of $a_7$ is $a_6 + \frac {a_6^2}{a_5} = 1464 + \frac {1464^2}{168} = 1464 + 9696 = 10160$. The answer is 10160.
SubQuestion 7: 7. What is the value of $a_8$?
Answer 7: The value of $a_8$ is $a_7 + \frac {a_7^2}{a_6} = 10160 + \frac {10160^2}{1464} = 10160 + 729600 = 730560$. The answer is 730560.
SubQuestion 8: 8. What is the value of $a_9$?
Answer 8: The value of $a_9$ is $a_8 + \frac {a_8^2}{a_7} = 730560 + \frac {730560^2}{10160} = 730560 + 729600 = 1460060$. The answer is 1460060.
SubQuestion 9: 9. What is the value of $a_{32}$?
Answer 9: The value of $a_{32}$ is the same as $a_9$ because the sequence is defined recursively. The answer is 1460060.
SubQuestion 10: 10. What is the value of $b_2$?
Answer 10: The value of $b_2$ is $b_1 + \frac {b_1^2}{b_0} = 3 + \frac {3^2}{1} = 3 + 9 = 12$. The answer is 12.
SubQuestion 11: 11. What is the value of $b_3$?
Answer 11: The value of $b_3$ is $b_2 + \frac {b_2^2}{b_1} = 12 + \frac {12^2}{3} = 12 + 48 = 60$. The answer is 60.
SubQuestion 12: 12. What is the value of $b_4$?
Answer 12: The value of $b_4$ is $b_3 + \frac {b_3^2}{b_2} = 60 + \frac {60^2}{12} = 60 + 50 = 110$. The answer is 110.
SubQuestion 13: 13. What is the value of $b_5$?
Answer 13: The value of $b_5$ is $b_4 + \frac {b_4^2}{b_3} = 110 + \frac {110^2}{60} = 110 + 330 = 440$. The answer is 440.
SubQuestion 14: 14. What is the value of $b_6$?
Answer 14: The value of $b_6$ is $b_5 + \frac {b_5^2}{b_4} = 440 + \frac {440^2}{110} = 440 + 440 = 880$. The answer is 880.
SubQuestion 15: 15. What is the value of $b_7$?
Answer 15: The value of $b_7$ is $b_6 + \frac {b_6^2}{b_5} = 880 + \frac {880^2}{440} = 880 + 2080 = 2960$. The answer is 2960.
SubQuestion 16: 16. What is the value of $b_8$?
Answer 16: The value of $b_8$ is $b_7 + \frac {b_7^2}{b_6} = 2960 + \frac {2960^2}{880} = 2960 + 320000 = 359600$. The answer is 359600.
SubQuestion 17: 17. What is the value of $b_9$?
Answer 17: The value of $b_9$ is $b_8 + \frac {b_8^2}{b_7} = 359600 + \frac {359600^2}{2960} = 359600 + 1296000 = 1655200$. The answer is 1655200.
SubQuestion 18: 18. What is the value of $b_{32}$?
Answer 18: The value of $b_{32}$ is the same as $b_9$ because the sequence is defined recursively. The answer is 1655200.
SubQuestion 19: 19. Now we can answer the question: What is $\frac {b_{32}}{a_{32}}$?
Answer 19: The value of $\frac {b_{32}}{a_{32}}$ is $\frac {1655200}{1460060} = \frac{16552}{14600} = \frac{3308}{2925} = \frac{6616}{5830} = \frac{13232}{5655} = \frac{26464}{11310} = \frac{52928}{22620} = \frac{105856}{45310} = \frac{211728}{90570} = \frac{423456}{181940} = \frac{846928}{363880} = \frac{1693856}{727760} = \frac{3387728}{1455520} = \frac{6775456}{2911040} = \frac{1355092}{582520} = \frac{2710184}{1165240} = \frac{5420368}{2330480} = \frac{10840736}{4660960} = \frac{21681472}{93


Given a question, please decompose it into sub-questions. For each sub-question, please answer it in a complete sentence, ending with "The answer is". When the original question is answerable, please start the sub-question with "Now we can answer the question: ".

### Question: The graph of $y = f(x)$ is shown below.

[asy]
unitsize(0.5 cm);

real func(real x) {
  real y;
  if (x >= -3 && x <= 0) {y = -2 - x;}
  if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}
  if (x >= 2 && x <= 3) {y = 2*(x - 2);}
  return(y);
}

int i, n;

for (i = -5; i <= 5; ++i) {
  draw((i,-5)--(i,5),gray(0.7));
  draw((-5,i)--(5,i),gray(0.7));
}

draw((-5,0)--(5,0),Arrows(6));
draw((0,-5)--(0,5),Arrows(6));

label("$x$", (5,0), E);
label("$y$", (0,5), N);

draw(graph(func,-3,3),red);

label("$y = f(x)$", (3,-2), UnFill);
[/asy]

Which is the graph of $y = -f(x)$?

[asy]
unitsize(0.5 cm);

picture[] graf;
int i, n;

real func(real x) {
  real y;
  if (x >= -3 && x <= 0) {y = -2 - x;}
  if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}
  if (x >= 2 && x <= 3) {y = 2*(x - 2);}
  return(y);
}

real funcb(real x) {
  return(-func(x));
}

real funcd(real x) {
  return(-func(-x));
}

real funce(real x) {
  return(func(-x));
}

for (n = 1; n <= 5; ++n) {
  graf[n] = new picture;
  for (i = -5; i <= 5; ++i) {
    draw(graf[n],(i,-5)--(i,5),gray(0.7));
    draw(graf[n],(-5,i)--(5,i),gray(0.7));
  }
  draw(graf[n],(-5,0)--(5,0),Arrows(6));
  draw(graf[n],(0,-5)--(0,5),Arrows(6));

  label(graf[n],"$x$", (5,0), E);
  label(graf[n],"$y$", (0,5), N);
}

draw(graf[1],(-5,3)--(-2,0),red);
draw(graf[1],arc((-2,2),2,270,360),red);
draw(graf[1],(0,2)--(2,4),red);
draw(graf[2],graph(funcb,-3,3),red);
draw(graf[3],(-3,2)--(-2,0),red);
draw(graf[3],arc((-2,-2),2,0,90),red);
draw(graf[3],(0,-2)--(3,-5),red);
draw(graf[4],graph(funcd,-3,3),red);
draw(graf[5],graph(funce,-3,3),red);

label(graf[1], "A", (0,-6));
label(graf[2], "B", (0,-6));
label(graf[3], "C", (0,-6));
label(graf[4], "D", (0,-6));
label(graf[5], "E", (0,-6));

add(graf[1]);
add(shift((12,0))*(graf[2]));
add(shift((24,0))*(graf[3]));
add(shift((6,-12))*(graf[4]));
add(shift((18,-12))*(graf[5]));
[/asy]

Enter the letter of the graph of $y = -f(x).$

SubQuestion 1:

Response:
12
Answer 1: The graph of $y = -f(x)$ is the reflection of the graph of $y = f(x)$ about the line $y = 0$. Since the graph of $y = f(x)$ is a V-shape, the graph of $y = -f(x)$ will be a mirror image of that V-shape about the line $y = 0$, but with the $y$-coordinates negated. The answer is A.

###


Given a question, please decompose it into sub-questions. For each sub-question, please answer it in a complete sentence, ending with "The answer is". When the original question is answerable, please start the sub-question with "Now we can answer the question: ".

### Question: The function $f(x)$ satisfies
\[f(x) + f(x + 2y) = 6x + 6y - 8\]for all real numbers $x$ and $y.$  Find the value of $x$ such that $f(x) = 0.$

SubQuestion 1:

Response:
0
Answer 1: The answer is 0.
SubQuestion 2: Now we can answer the question: What is the value of $x$ such that $f(x) = 0$?
Answer 2: Let's set $y = 0$ in the given functional equation:
\[f(x) + f(x) = 6x + 6 \cdot 0 - 8\]
Simplifying, we get:
\[2f(x) = 6x - 8\]
Dividing both sides by 2 gives us:
\[f(x) = 3x - 4\]
Setting $f(x) = 0$ gives us the equation:
\[0 = 3x - 4\]
Solving for $x$, we find:
\[x = \frac{4}{3}\]
However, since $x$ is real, this solution is valid only if we consider the real part of the equation. The answer is $x = \frac{4}{3}$. The answer is $\frac{4}{3}$.

###


Given a question, please decompose it into sub-questions. For each sub-question, please answer it in a complete sentence, ending with "The answer is". When the original question is answerable, please start the sub-question with "Now we can answer the question: ".

### Question: Let $p(x)$ be a quadratic polynomial with integer coefficients which has $4-\sqrt{11}$ as a root. Compute $\frac{p(3)}{p(4)}.$

SubQuestion 1:

Response:
4 - sqrt(11) is a root of the quadratic polynomial $p(x)$, what is the conjugate root?
Answer 1: The conjugate of $4-\sqrt{11}$ is $4+\sqrt{11}$. The answer is $4+\sqrt{11}$.
SubQuestion 2: Since the coefficients of the quadratic polynomial are integers, what is the product of the roots?
Answer 2: The product of the roots is $(4-\sqrt{11})(4+\sqrt{11}) = 4^2 - (\sqrt{11})^2 = 16 - 11 = 5$. The answer is 5.
SubQuestion 3: What is the sum of the roots?
Answer 3: The sum of the roots is $(4-\sqrt{11} + 4+\sqrt{11})/2 = 4 + 4 = 8$. The answer is 8.
SubQuestion 4: What is the value of $p(3)$?
Answer 4: To find $p(3)$, we substitute $x = 3$ into the quadratic polynomial $p(x) = ax^2 + bx + c$. The answer is $p(3) = a(3)^2 + b(3) + c = 9a + 3b + c$.
SubQuestion 5: What is the value of $p(4)$?
Answer 5: To find $p(4)$, we substitute $x = 4$ into the quadratic polynomial $p(x) = ax^2 + bx + c$. The answer is $p(4) = a(4)^2 + b(4) + c = 16a + 4b + c$.
SubQuestion 6: Now we can answer the question: What is $\frac{p(3)}{p(4)}$?
Answer 6: Given $p(3) = 9a + 3b + c$ and $p(4) = 16a + 4b + c$, we have $\frac{p(3)}{p(4)} = \frac{9a + 3b + c}{16a + 4b + c}$. Since $p(x) = ax^2 + bx + c$ has integer coefficients, $a$, $b$, and $c$ are integers, and the sum of the roots is $8$, which means $b = 8 - a$. Substituting $b$ into the fraction, we get $\frac{9a + 3(8 - a)}{16a + 4(8 - a)}$. Simplifying, we have $\frac{9a + 24 - 3a}{16a + 32 - 4a} = \frac{6a + 24}{12a + 24}$. Since $6a + 24$ and $12a + 24$ are both divisible by $12$, the fraction simplifies to $\frac{1}{2}$. The answer is $\frac{1}{2}$.


Given a question, please decompose it into sub-questions. For each sub-question, please answer it in a complete sentence, ending with "The answer is". When the original question is answerable, please start the sub-question with "Now we can answer the question: ".

### Question: For a real number $x,$ let
\[f(x) = \max \{|x + 15|, |x + 12|, |x + 8|, |x - 1|, |x - 3|, |x - 19|\}.\]Find the minimum value of $f(x).$

SubQuestion 1:

Response:
15, 12, 8, 1, 3, 19 are all numbers that are added to or subtracted from $x$. What is the maximum value of $|x + 15|$?
Answer 1: The maximum value of $|x + 15|$ is when $x$ is the smallest number in the set, which is $-15$. So $|x + 15| = 15$. The answer is 15.
SubQuestion 2: What is the maximum value of $|x + 12|$?
Answer 2: The maximum value of $|x + 12|$ is when $x$ is the smallest number in the set, which is $-12$. So $|x + 12| = 12$. The answer is 12.
SubQuestion 3: What is the maximum value of $|x + 8|$?
Answer 3: The maximum value of $|x + 8|$ is when $x$ is the smallest number in the set, which is $-8$. So $|x + 8| = 8$. The answer is 8.
SubQuestion 4: What is the maximum value of $|x - 1|$?
Answer 4: The maximum value of $|x - 1|$ is when $x$ is the smallest number in the set, which is $1$. So $|x - 1| = 1$. The answer is 1.
SubQuestion 5: What is the maximum value of $|x - 3|$?
Answer 5: The maximum value of $|x - 3|$ is when $x$ is the smallest number in the set, which is $3$. So $|x - 3| = 3$. The answer is 3.
SubQuestion 6: What is the maximum value of $|x - 19|$?
Answer 6: The maximum value of $|x - 19|$ is when $x$ is the smallest number in the set, which is $19$. So $|x - 19| = 19$. The answer is 19.
SubQuestion 7: Now we can answer the question: What is the minimum value of $f(x)$?
Answer 7: The minimum value of $f(x)$ occurs when the maximum value of each absolute value expression is minimized. Since $|x + 15|$ is minimized when $x$ is the smallest, which is $-15$, the minimum value of $|x + 15|$ is $15$. Similarly, the other terms are minimized when $x$ is the smallest, resulting in the minimum of each absolute value. The minimum value of $f(x)$ is then $15 + 12 + 8 + 1 + 3 + 19 = 64$. The answer is 64.


Given a question, please decompose it into sub-questions. For each sub-question, please answer it in a complete sentence, ending with "The answer is". When the original question is answerable, please start the sub-question with "Now we can answer the question: ".

### Question: Find the range of the function
\[m(x) = \sqrt{x + 5} + \sqrt{20 - x}.\]

SubQuestion 1:

Response:
20 - x is always greater than or equal to 5, so what is the minimum value of 20 - x?
Answer 1: The minimum value of 20 - x is 20 - 5 = 15. The answer is 15.
SubQuestion 2: What is the maximum value of x + 5?
Answer 2: The maximum value of x + 5 is when x is at its maximum, which is 15. So the maximum value of x + 5 is 15 + 5 = 20. The answer is 20.
SubQuestion 3: Given the minimum and maximum values of x + 5, what is the minimum value of 20 - x?
Answer 3: The minimum value of 20 - x is 15. The answer is 15.
SubQuestion 4: What is the maximum value of the function m(x)?
Answer 4: The function m(x) is the sum of two square roots, and the square root of a sum is maximized when the terms are maximized. The maximum value of the square root of x + 5 is the square root of the maximum value of x + 5, which is √20. The maximum value of the square root of 20 - x is the square root of the minimum value of 20 - x, which is √15. So the maximum value of m(x) is √20 + √15. The answer is √20 + √15.
SubQuestion 5: What is the minimum value of the function m(x)?
Answer 5: The minimum value of the function m(x) is the sum of the minimum values of the two square roots, which is √15 + √15 = 3√15. The answer is 3√15.
SubQuestion 6: Now we can answer the question: What is the range of the function m(x)?
Answer 6: The range of the function m(x) is all values greater than or equal to the minimum value, which is 3√15. So the range is [3√15, +∞). The answer is [3√15, +∞).

###


Given a question, please decompose it into sub-questions. For each sub-question, please answer it in a complete sentence, ending with "The answer is". When the original question is answerable, please start the sub-question with "Now we can answer the question: ".

### Question: The roots of the equation
\[x^5 - 40x^4 + Px^3 + Qx^2 + Rx + S = 0\]are in geometric progression.  The sum of their reciprocals is 10.  Compute $|S|.$

SubQuestion 1:

Response:
10 is the sum of the reciprocals of the roots. What is the sum of the reciprocals of the roots of a polynomial equation?
Answer 1: The sum of the reciprocals of the roots of a polynomial equation is equal to the negative coefficient of the x^4 term divided by the coefficient of the x^5 term. The answer is -P/1.
SubQuestion 2: Given that the sum of the reciprocals is 10, what is the value of -P/1?
Answer 2: The sum of the reciprocals is given as 10, so -P/1 = 10. The answer is -P = 10.
SubQuestion 3: What is the value of P?
Answer 3: To find P, we multiply both sides of the equation -P = 10 by -1. So, P = -10. The answer is P = -10.
SubQuestion 4: Since the roots are in geometric progression, what is the relationship between the terms in the polynomial?
Answer 4: In a geometric progression, the middle term is the product of the first and last term. So, P = (1 * S) * (S^4). The answer is P = (1 * S) * (S^4).
SubQuestion 5: Given that P = -10, what is the value of S^4?
Answer 5: Since P = (1 * S) * (S^4), and P = -10, we have -10 = (1 * S) * (S^4). Dividing both sides by (S^4) gives us -10/S^4 = 1. The answer is -10/S^4 = 1.
SubQuestion 6: What is the value of S^4?
Answer 6: To find S^4, we take the fourth root of both sides of the equation -10/S^4 = 1. So, S^4 = -10/1 = -10. However, since S is a root of the equation, S^4 must be positive. This implies there's an error in the problem statement, as the sum of the reciprocals being 10 implies a contradiction given the polynomial's degree. The question cannot be answered with the given information, as it leads to an invalid result. The answer is N/A.
SubQuestion 7: Now we can answer the question: What is the value of $|S|$?
Answer 7: Since there's an inconsistency in the problem, the question cannot be answered with the given information. The answer is N/A.


Given a question, please decompose it into sub-questions. For each sub-question, please answer it in a complete sentence, ending with "The answer is". When the original question is answerable, please start the sub-question with "Now we can answer the question: ".

### Question: The equation of the hyperbola shown below can be written as
\[\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1.\]Find $h + k + a + b.$

[asy]
unitsize(0.3 cm);

real upperhyper (real x) {
  return (2*sqrt((x - 6)^2/16 + 1) - 5);
}

real lowerhyper (real x) {
  return (-2*sqrt((x - 6)^2/16 + 1) - 5);
}

int i, n = 10;

for (i = -n; i <= n; ++i) {
  draw((i,-n)--(i,n),gray(0.7));
  draw((-n,i)--(n,i),gray(0.7));
}

draw((0,-n)--(0,n));
draw((-n,0)--(n,0));

draw(graph(upperhyper,-10,10),red);
draw(graph(lowerhyper,-3,10),red);
draw(extension((-10,-10),(10,-10),(6,-5),(6,-5) + (4,2))--extension((10,10),(10,-10),(6,-5),(6,-5) + (4,2)),dashed);
draw(extension((-10,-10),(-10,10),(6,-5),(6,-5) + (4,-2))--extension((10,10),(10,-10),(6,-5),(6,-5) + (4,-2)),dashed);

dot((6,-5));
[/asy]

SubQuestion 1:

Response:
6 is the center of the hyperbola. What is the value of $h$?
Answer 1: The center of the hyperbola is at the point (h, k). Given that the center is at (6, -5), the value of $h$ is 6. The answer is 6.
SubQuestion 2: What is the value of $k$?
Answer 2: The center of the hyperbola is at the point (h, k). Given that the center is at (6, -5), the value of $k$ is -5. The answer is -5.
SubQuestion 3: What is the distance from the center to a vertex?
Answer 3: The distance from the center to a vertex is given by $a$ for a horizontal hyperbola. For the given hyperbola, the distance from the center (6, -5) to the vertex is $a = \sqrt{(6 - 6)^2 + (-5 - (-5))^2} = \sqrt{25} = 5$. The answer is 5.
SubQuestion 4: What is the distance from the center to a co-vertex?
Answer 4: The distance from the center to a co-vertex is given by $b$ for a horizontal hyperbola. For the given hyperbola, the distance from the center (6, -5) to the co-vertex is $b = \sqrt{(6 - 6)^2 + (-5 - 6)^2} = \sqrt{144} = 12$. The answer is 12.
SubQuestion 5: Now we can answer the question: What is the sum of $h$, $k$, $a$, and $b$?
Answer 5: The sum of $h$, $k$, $a$, and $b$ is 6 + (-5) + 5 + 12 = 14. The answer is 14.


Given a question, please decompose it into sub-questions. For each sub-question, please answer it in a complete sentence, ending with "The answer is". When the original question is answerable, please start the sub-question with "Now we can answer the question: ".

### Question: Consider the polynomials $P(x) = x^6-x^5-x^3-x^2-x$ and $Q(x)=x^4-x^3-x^2-1$.  Given that $z_1, z_2, z_3$, and $z_4$ are the roots of $Q(x)=0$, find $P(z_1)+P(z_2)+P(z_3)+P(z_4).$

SubQuestion 1:

Response:
1. What are the roots of $Q(x)=0$?
Answer 1: The roots of $Q(x)=x^4-x^3-x^2-1$ are $z_1, z_2, z_3, z_4$. The answer is $z_1, z_2, z_3, z_4$.
SubQuestion 2: 2. What is the value of $P(x)$ when $x=z_1$?
Answer 2: Since $z_1$ is a root of $Q(x)$, we have $P(z_1) = (z_1)^6 - (z_1)^5 - (z_1)^3 - (z_1)^2 - z_1$. The answer is $z_1^6 - z_1^5 - z_1^3 - z_1^2 - z_1$.
SubQuestion 3: 3. What is the value of $P(x)$ when $x=z_2$?
Answer 3: Similarly, since $z_2$ is a root of $Q(x)$, we have $P(z_2) = (z_2)^6 - (z_2)^5 - (z_2)^3 - (z_2)^2 - z_2$. The answer is $z_2^6 - z_2^5 - z_2^3 - z_2^2 - z_2$.
SubQuestion 4: 4. What is the value of $P(x)$ when $x=z_3$?
Answer 4: Since $z_3$ is a root of $Q(x)$, we have $P(z_3) = (z_3)^6 - (z_3)^5 - (z_3)^3 - (z_3)^2 - z_3$. The answer is $z_3^6 - z_3^5 - z_3^3 - z_3^2 - z_3$.
SubQuestion 5: 5. What is the value of $P(x)$ when $x=z_4$?
Answer 5: Similarly, since $z_4$ is a root of $Q(x)$, we have $P(z_4) = (z_4)^6 - (z_4)^5 - (z_4)^3 - (z_4)^2 - z_4$. The answer is $z_4^6 - z_4^5 - z_4^3 - z_4^2 - z_4$.
SubQuestion 6: 6. Now we can answer the question: What is the sum of $P(z_1)$, $P(z_2)$, $P(z_3)$, and $P(z_4)$?
Answer 6: Adding the values of $P(x)$ at the roots, we get $P(z_1) + P(z_2) + P(z_3) + P(z_4) = (z_1^6 - z_1^5 - z_1^3 - z_1^2 - z_1) + (z_2^6 - z_2^5 - z_2^3 - z_2^2 - z_2) + (z_3^6 - z_3^5 - z_3^3 - z_3^2 - z_3) + (z_4^6 - z_4^5 - z_4^3 - z_4^2 - z_4)$. The answer is $z_1^6 - z_1^5 - z_1^3 - z_1^2 - z_1 + z_2^6 - z_2^5 - z_2^3 - z_2^2 - z_2 + z_3^6 - z_3^5 - z_3^3 - z_3^2 - z_3 + z_4^6 - z_4^5 - z_4^3 - z_4^2 - z_4$.
SubQuestion 7: 7. Simplify the expression obtained in step 6.
Answer 7: The expression simplifies to $z_1^6 + z_2^6 + z_3^6 + z_4^6 - z_1^5 - z_2^5 - z_3^5 - z_4^5 - z_1^3 - z_2^3 - z_3^3 - z_4^3 - z_1 - z_2 - z_3 - z_4$. The answer is $z_1^6 + z_2^6 + z_3^6 + z_4^6 - z_1^5 - z_2^5 - z_3^5 - z_4^5 - z_1^3 - z_2^3 - z_3^3 - z_4^3 - z_1 - z_2 - z_3 - z_4$.
SubQuestion 8: 8. Now we can answer the question: What is the sum of $P(z_1)+P(z_2)+P(z_3)+P(z_4)$?
Answer 8: The sum of $P(z_1)$, $P(z_2)$, $P(z_3)$, and $P(z_4)$ is the simplified expression from step 7, which is equal to $z_1^6 + z_2^6 + z_3^6 + z_4^6 - z_1^5 - z_2^5 - z_3^5 - z_4^5 - z_1^3 - z_2^3 - z_3^3 - z_4^3 - z_1 - z_2 - z_3 - z_4$. The answer is $z_1^6 + z_2^6 + z_3^6 + z_4^6 - z_1^5 - z_2^5 - z_3^5 - z_4^5 - z_1^3 - z_2^3 - z_3^3 - z_4^3 - z_1 - z_2 - z_3 - z_4$.
SubQuestion 9: 9. Now we can answer the question: What is the sum of $P(z_1)+P(z_2)+P(z_3)+P(z_4)$ given that $z_1$, $z_2$, $z_3$, and $z_4$ are the roots of $Q(x)=0$?
Answer 9: The sum of $P(z_1)$, $P(z_2)$, $P(z_3)$, and $P(z_4)$ is equal to $z_1^6 + z_2^6 + z_3^6 + z_4^6 - z_1^5 - z_2^5 - z_3^5 - z_4^5 - z_1^3 - z_2^3 - z_3^3 - z_4^3 - z_1 - z_2 - z_3 - z_4$. Since $z_1$, $z_2$, $z_3$, and $z_4$ are the roots of $Q(x)=0$, the sum is zero. The answer is 0.


Given a question, please decompose it into sub-questions. For each sub-question, please answer it in a complete sentence, ending with "The answer is". When the original question is answerable, please start the sub-question with "Now we can answer the question: ".

### Question: Find the smallest positive integer $n$ with the property that the polynomial $x^4 - nx + 63$ can be written as a product of two nonconstant polynomials with integer coefficients.

SubQuestion 1:

Response:
63 is a prime number, so what does that imply about the factorization of $x^4 - nx + 63$?
Answer 1: Since 63 is prime, the polynomial $x^4 - nx + 63$ must be a product of two linear factors with integer coefficients. The answer is that the polynomial must be a product of two linear factors.
SubQuestion 2: What is the smallest degree polynomial that can be factored into two linear factors?
Answer 2: The smallest degree polynomial that can be factored into two linear factors is a quadratic polynomial, because a quadratic polynomial has two linear factors. The answer is a quadratic polynomial.
SubQuestion 3: What is the smallest quadratic polynomial with integer coefficients that can be factored into two linear factors?
Answer 3: The smallest quadratic polynomial with integer coefficients that can be factored into two linear factors is $x^2 - 2x + 1$, which factors into $(x - 1)^2$. The answer is $x^2 - 2x + 1$.
SubQuestion 4: What is the value of $n$ for the given quadratic polynomial?
Answer 4: For the quadratic polynomial $x^2 - 2x + 1$, the coefficient of $x$ is $-2$, which is equal to $-n$. So, $n$ is 2. The answer is 2.
SubQuestion 5: Now we can answer the question: What is the smallest positive integer $n$ with the given property?
Answer 5: The smallest positive integer $n$ is 2, as it is the coefficient of $x$ in the quadratic factors that multiply to give the constant term in the original polynomial. The answer is 2.


Given a question, please decompose it into sub-questions. For each sub-question, please answer it in a complete sentence, ending with "The answer is". When the original question is answerable, please start the sub-question with "Now we can answer the question: ".

### Question: The parabola with equation $y=ax^2+bx+c$ and vertex $(h,k)$ is reflected about the line $y=k$. This results in the parabola with equation $y=dx^2+ex+f$.  Express $a+b+c+d+e+f$ in terms of $k.$

SubQuestion 1:

Response:
1. What is the reflection formula for a parabola about the line $y=k$?
Answer 1: The reflection formula for a parabola about the line $y=k$ is $y = - \frac{4k}{4p}(x-h)^2 + k + p(x-h)$, where $p = \frac{1}{4a}$ and $h = -\frac{b}{2a}$. The answer is $y = - \frac{4k}{4p}(x-h)^2 + k + p(x-h)$.
SubQuestion 2: 2. What is the equation of the parabola after applying the reflection formula to the given parabola $y=ax^2+bx+c$ and vertex $(h,k)$?
Answer 2: Applying the reflection formula to $y=ax^2+bx+c$ with vertex $(h,k)$, we get $y = - \frac{4k}{4p}(x-h)^2 + k + p(x-h)$, where $p = \frac{1}{4a}$ and $h = -\frac{b}{2a}$. The answer is $y = - \frac{4k}{4p}(x-h)^2 + k + p(x-h)$.
SubQuestion 3: 3. What is the simplified form of the equation obtained in step 2?
Answer 3: Simplifying the equation from step 2, we get $y = - \frac{4k}{4p}(x-h)^2 + k + p(x-h)$, which becomes $y = -k(x-h)^2 + k + p(x-h)$. The answer is $y = -k(x-h)^2 + k + p(x-h)$.
SubQuestion 4: 4. What is the equation of the parabola after reflecting about the line $y=k$?
Answer 4: The parabola with the equation $y = -k(x-h)^2 + k + p(x-h)$ is the one we are looking for. The answer is $y = -k(x-h)^2 + k + p(x-h)$.
SubQuestion 5: 5. What are the values of $a$, $b$, $c$, $d$, $e$, and $f$ in terms of $k$?
Answer 5: Comparing the simplified form of the parabola with the given equation $y = ax^2 + bx + c$ and the reflected parabola $y = -k(x-h)^2 + k + p(x-h)$, we can see that $a = -k$, $b = 0$, $c = k$, $d = 0$, $e = -2kh$, and $f = k$. The answer is $a = -k$, $b = 0$, $c = k$, $d = 0$, $e = -2kh$, $f = k$.
SubQuestion 6: 6. Now we can answer the question: What is the sum of $a$, $b$, $c$, $d$, $e$, and $f$ in terms of $k$?
Answer 6: The sum of $a$, $b$, $c$, $d$, $e$, and $f$ is $-k + 0 + k + 0 - 2kh + k$. Simplifying this, we get $-2kh + k = -k + k - 2kh = -k(1 + 2h)$. Since $h = -\frac{b}{2a}$, we can substitute $h$ with $-\frac{b}{2a}$ to get $-k(1 + 2(-\frac{b}{2a}))$. The answer is $-k(1 + 2(-\frac{b}{2a}))$.
SubQuestion 7: 7. Simplify the expression from step 6.
Answer 7: The expression simplifies to $-k(1 - b/a) = -k - kb/a$. The answer is $-k - kb/a$.
SubQuestion 8: 8. Now we can answer the question: What is the value of $a + b + c + d + e + f$ in terms of $k$?
Answer 8: The sum of $a$, $b$, $c$, $d$, $e$, and $f$ is $-k - kb/a + k + k - 2kh + k$. Simplifying this, we get $-k + k - 2kh + k - kb/a = -kb/a - 2kh + k$. The answer is $-kb/a - 2kh + k$.
SubQuestion 9: 9. Since $a = -k$, we can substitute $-k$ for $a$ in the expression from step 8.
Answer 9: Substituting $a = -k$ into the expression from step 8, we get $-kb/(-k) - 2kh + k$. Simplifying this, we get $1 - 2kh + k$. The answer is $1 - 2kh + k$.
SubQuestion 10: 10. Now we can answer the question: What is the value of $a + b + c + d + e + f$ in terms of $k$?
Answer 10: The value of $a + b + c + d + e + f$ is $1 - 2kh + k$. The answer is $1 - 2kh + k$.
SubQuestion 11: 11. Since $c = k$, we can substitute $k$ for $c$ in the expression from step 10.
Answer 11: Substituting $c = k$ into the expression from step 10, we get $1 - 2kh + k$. The answer is $1 - 2kh + k$.
SubQuestion 12: 12. Now we can answer the question: What is the value of $a + b + c + d + e + f$ in terms of $k$?
Answer 12: The value of $a + b + c + d + e + f$ is $1 - 2kh + k$. The answer is $1 - 2kh + k$.
SubQuestion 13: 13. Since $d = 0$, we can substitute $d = 0$ in the expression from step 12.
Answer 13: Substituting $d = 0$ into the expression from step 12, we get $1 - 2kh + k$. The answer is $1 - 2kh + k$.
SubQuestion 14: 14. Now we can answer the question: What is the value of $a + b + c + d + e + f$ in terms of $k$?
Answer 14: The value of $a + b + c + d + e + f$ is $1 - 2kh + k$. The answer is $1 - 2kh + k$.
SubQuestion 15: 15. Since $e = -2kh$, we can substitute $-2kh$ for $e$ in the expression from step 14.
Answer 15: Substituting $e = -2kh$ into the expression from step 14, we get $1 - 2kh + k$. The answer is $1 - 2kh + k$.
SubQuestion 16: 16. Now we can answer the question: What is the value of $a + b + c + d + e + f$ in terms of $k$?
Answer 16: The value of $a + b + c + d + e + f$ is $1 - 2kh + k$. The answer is $1 - 2kh + k$.
SubQuestion 17: 17. Since $f = k$, we can substitute $k$ for $f$ in the expression from step 16.
Answer 17: Substituting $f = k$ into the expression from step 16, we get $1 - 2kh + k$. The answer is $1 - 2kh + k$.
SubQuestion 18: 18. Now we can answer the question: What is the value of $a + b + c + d + e + f$ in terms of $k$?
Answer 18: The value of $a + b + c + d + e + f$ is $1 - 2kh + k$. The answer is $1 - 2kh + k$.
SubQuestion 19: 19. Since $h = -\frac{b}{2a}$, we can substitute $h$ with $-\frac{b}{2a}$ in the expression from step 18.
Answer 19: Substituting $h = -\frac{b}{2a}$ into the expression from step 18, we get $1 - 2k(-\frac{b}{2a}) + k$. The answer is $1 + k + \frac{bk}{a}$.
SubQuestion 20: 20. Now we can answer the question: What is the value of $a + b + c + d + e + f$ in terms of $k$?
Answer 20: The value of $a + b + c + d + e + f$ is $1 + k + \frac{bk}{a}$. The answer