inclusionAI/Ring-lite-rl-data huggingface dataset:
Adam owns a square $S$ with side length $42$. First, he places rectangle $A$, which is $6$ times as long as it is wide, inside the square, so that all four vertices of $A$ lie on sides of $S$, but none of the sides of $ A$ are parallel to any side of $S$. He then places another rectangle $B$, which is $ 7$ times as long as it is wide, inside rectangle $A$, so that all four vertices of $ B$ lie on sides of $ A$, and again none of the sides of $B$ are parallel to any side of $A$. Find the length of the shortest side of rectangle $ B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{29}{4}$}
Cindy has a collection of identical rectangular prisms. She stacks them, end to end, to form $1$ longer rectangular prism. Suppose that joining $11$ of them will form a rectangular prism with $3$ times the surface area of an individual rectangular prism. How many will she need to join end to end to form a rectangular prism with $9$ times the surface area?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Find the distribution of the random variable $Y = X^2$, where $X$ is a standard normal random variable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\chi^2(1)}
What is the value of $\left(\left((3+2)^{-1}-1\right)^{-1}-1\right)^{-1}-1$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{13}{9}}
Determine the big-O bound for the following expression:
\[ \sum^{ N}_{k=0}{2^k \log_2\left(\frac{2^N}{2^k}\right)} \]
where \( N = \log M \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(M)}
Find the sum of the eccentric angles \(\alpha, \beta, \gamma,\) and \(\delta\) of four concyclic points on the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
For how many integers $1\leq n \leq 2010$, $2010$ divides $1^2-2^2+3^2-4^2+\dots+(2n-1)^2-(2n)^2$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
8
}
Suppose you draw random variables $x_i$ from the interval $[0,1]$, where each $x_i$ is uniformly distributed. If $x_i > x_{i-1}$, you draw again; otherwise, you stop. What is the expected number of draws before the sequence stops?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e}
Find a closed-form expression that bounds the sum \( \sum_{i=1}^{\log^* N} \log^{(i)}N \), where \( \log^* \) denotes the iterated logarithm function and \( \log^{(i)}N \) represents the logarithm function applied iteratively \( i \) times.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \log N}
Solve the complex equation \((z+1)^4 = (\bar{z}+i)^4\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{z = t(1 - i)}
3. The base of the pyramid $P A B C D$ is a trapezoid $A B C D$, the base $A D$ of which is twice the length of the base $B C$. The segment $M N$ is the midline of the triangle $A B P$, parallel to the side $A B$. Find the ratio of the volumes of the solids into which the plane $D M N$ divides this pyramid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{13}}
Find a space that is homeomorphic to the quotient space $\mathbb{R}^3 / \sim$, where the equivalence relation $\sim$ is defined by $(x, y, z) \sim (x, y', z')$ for any $y', z' \in \mathbb{R}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}}
Find the largest value of the real number $ \lambda$, such that as long as point $ P$ lies in the acute triangle $ ABC$ satisfying $ \angle{PAB}=\angle{PBC}=\angle{PCA}$, and rays $ AP$, $ BP$, $ CP$ intersect the circumcircle of triangles $ PBC$, $ PCA$, $ PAB$ at points $ A_1$, $ B_1$, $ C_1$ respectively, then $ S_{A_1BC}+ S_{B_1CA}+ S_{C_1AB} \geq \lambda S_{ABC}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\[ \lambda = 3 \]}
Evaluate the limit of the series as \( x \to \infty \):
\[ \lim_{x \rightarrow \infty} \sum_{n=0}^\infty (-1)^{n} \frac{x \lvert x \rvert^{2n}}{2n+1} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
Given a parabola \( y^2 = 6x \) with two variable points \( A(x_1, y_1) \) and \( B(x_2, y_2) \), where \( x_1 \neq x_2 \) and \( x_1 + x_2 = 4 \). The perpendicular bisector of segment \( AB \) intersects the x-axis at point \( C \). Find the maximum area of triangle \( \triangle ABC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{14}{3}\sqrt{7}}
1. Evan's analog clock displays the time $12: 13$; the number of seconds is not shown. After 10 seconds elapse, it is still $12: 13$. What is the expected number of seconds until $12: 14$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Determine the smallest number of subsets, $n$, into which the set $S = \{1, 2, \dots, 2022\}$ can be partitioned such that for each subset $S_i$, either all pairs of distinct elements have a GCD greater than 1, or all pairs of distinct elements have a GCD equal to 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Let $S$ be the set of all points $(x,y)$ in the coordinate plane such that $0\leq x\leq \frac{\pi}{2}$ and $0\leq y\leq
\frac{\pi}{2}$. What is the area of the subset of $S$ for which \[
\sin^2 x -\sin x \sin y +\sin^2 y \leq\frac{3}{4}\,?
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi^2}{6}}
The numbers 1, 2, ..., 2002 are written in order on a blackboard. Then the 1st, 4th, 7th, ..., 3k+1th, ... numbers in the list are erased. The process is repeated on the remaining list (i.e., erase the 1st, 4th, 7th, ... 3k+1th numbers in the new list). This continues until no numbers are left. What is the last number to be erased?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1598}
Determine the number of distinct real roots of the equation:
\[ \sum\limits_{i = 1}^{40}|a_i - x| = \sum\limits_{i = 1}^{40}|b_i - x|. \]
Here, \(a_i\) and \(b_i\) are real numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
In a right triangle ABC with \(\angle B=90^\circ\), two equilateral triangles ABD and BEC are drawn externally on the legs of triangle ABC. Let G, H, and F be the midpoints of BE, BC, and DC, respectively. If the area of triangle ABC is 32, find the area of triangle GHF.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let $F_0 = 0$, $F_1 = 1$, and for $n \ge 1$, $F_{n+1} = F_n + F_{n-1}$. Define $a_n = \left(\frac{1 + \sqrt{5}}{2}\right)^n \cdot F_n$. Then there are rational numbers $A$ and $B$ such that:

$$\frac{a_{30} + a_{29}}{a_{26} + a_{25}} = A + B \sqrt{5}.$$ 

Find $A + B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
Evaluate the iterated integral \( \iint f \, \mathrm{d}\nu \, \mathrm{d}\mu \) for the function \( f(m,n) \) defined as follows: \( f(m,n) = 1 \) if \( m = n \), \( f(m,n) = -1 \) if \( m = n+1 \), and \( f(m,n) = 0 \) otherwise, where \( \mu = \nu \) are counting measures on \( \mathbb{N} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Problem 4.1. Along the road connecting Masha's and Sasha's houses, there are 17 apple trees and 18 poplars. When Masha was going to visit Sasha, she took photos of all the trees. Right after the tenth apple tree, Masha's phone memory ran out, and she couldn't photograph the remaining 13 trees. The next day, when Sasha was going to visit Masha, starting from the eighth apple tree, he picked one leaf from each tree. How many leaves did Sasha pick?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
In a knockout tournament with 55 boxers, each match is held sequentially. In every match, the difference in the number of previous victories between the two participants is at most 1. What is the maximum number of matches the tournament winner could have fought?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with $a > b > 0$, the right vertex is $A(a, 0)$. There exists a point $P$ on the ellipse such that $\angle APO = 90^{\circ}$. Determine the range of values for the eccentricity of the ellipse.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left( \frac{\sqrt{2}}{2}, 1 \right)}
Find the range of the expression $\frac{a \cos x - b \sin x + 2c}{\sqrt{a^2 + b^2 + c^2}}$ where $x, a, b,$ and $c$ are arbitrary numbers such that $a^2 + b^2 + c^2 \neq 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-\sqrt{5}, \sqrt{5}]}
Children are assembling robot toys. Initially, every 2 children cooperate to assemble 1 small robot toy, then every 3 children cooperate to assemble 1 large robot toy. A total of 30 robot toys of various sizes were assembled. How many children are there in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
Calculate the sum of all four-digit numbers that can be formed using the digits 0, 1, 2, 3, and 4, with no repeated digits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{259980}
If real numbers \(x\) and \(y\) satisfy \(x^2 + y^2 = 20\), what is the maximum value of \(xy + 8x + y\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Find the closed form of the following series: $$\sum_{r\ge 0} \binom{n}{r}\binom{n-r}{r} 2^{n-2r}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\binom{2n}{n}}
Jennifer plans a profit of 20% on the selling price of an item, and her expenses are 10% of the selling price. There is also a sales tax of 5% on the selling price of the item. The item sells for $10.00. Calculate the rate of markup on cost of this item.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{53.85\%}
Seven old women are going to Rome. Each has seven mules, each mule carries seven bags, each bag contains seven loaves of bread, each loaf contains seven knives, each knife is in seven sheaths. How many total items are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{137256}
Tim and Allen are playing a match of *tenus*.  In a match of *tenus*, the two players play a series of games, each of which is won by one of the two players. The match ends when one player has won exactly two more games than the other player, at which point the player who has won more games wins the match. In odd-numbered games, Tim wins with probability  $3/4$ , and in the even-numbered games, Allen wins with probability  $3/4$ .  What is the expected number of games in a match?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{16}{3}}
An amount is invested in an account that earns 6% interest compounded semi-annually. The interest earned after one year is $6 greater than if the same amount had been invested at 8% simple yearly interest. What is the amount invested?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$314.14$}
A group of people are playing paintball. In a given situation, all distances between individuals are different. Everyone shoots at the person closest to them. What is the maximum number of people that can shoot at the same person?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find the number of sequences of 10 letters where all the letters are either $A$ or $B$, the first letter is $A$, the last letter is $B$, and the sequence contains no three consecutive letters reading $ABA$. For example, count $AAABBABBAB$ and $ABBBBBBBAB$ but not $AABBAABABB$ or $AAAABBBBBA$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{86}
X is the number of 2 groups of 10 and 6 units, and Y is the number of 4 groups of 10 and 1 unit. Find the sum of X and Y.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
We drew 11 horizontal and 11 vertical intersecting straight lines on a sheet of paper. We call a segment of one of the straight line drawn that connects two neighboring intersections a “link”. What least number of links must we erase in order for each intersection to be a junction of not more than 3 links?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41}
设取整 $\left[\frac{10^{2022}}{10^{100}-9}\right]$ 被 $10^{n}$ 整除, 则非负整数 n的最大可能值为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Determine the number of ring homomorphisms from $\mathbb{Z}_{12}$ into $\mathbb{Z}_{24}$ using the formula provided.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
What is the smallest number which when increased by a certain number is divisible by 27, 35, 25, & 21 and the result is 4728?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4725}
24. Use red, pink, yellow, and black 4 colors to paint the 6 sides of a regular hexagon, so that the sides painted yellow and black each have one, and the sides painted red and pink each have 2. Mr. Li has some color blindness. For any two colored regular hexagons $A_{6}$ and $A_{6}^{\prime}$, if $A_{6}^{\prime}$ is obtained by changing the two sides painted red in $A_{6}$ to pink, and the two sides painted pink to red, Mr. Li, due to his color blindness, considers $A_{6}$ and $A_{6}^{\prime}$ to be the same. Find the number of distinct colorings of the hexagon under Mr. Li's identification method.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
10. In the Cartesian coordinate system $x O y$, it is known that the vertices of the equilateral $\triangle A B C$ (vertices in counterclockwise order) are $A(-2,2)$, vertex $B$ is on the $y$-axis, and $C B \parallel x$-axis; point $D(2,6)$, $E(2,2)$, and the radius of $\odot O$ is 2. Find the radius of the circle that is externally tangent to $\odot O$ and tangent to line segments $A B$ and $D E$ (accurate to $10^{-4}$).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.4726}
The racing magic takes some time to circle the racing track once. The charging bull makes 40 rounds of the track in an hour. If they left the starting point together, it took them 6 minutes to meet at the starting point for the second time. How many seconds does it take for the racing magic to circle the track once?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{360}
A parallelogram contains two circles. One of them, with a radius of 3, is inscribed in the parallelogram, while the second circle touches two sides of the parallelogram and the first circle. The distance between the points of tangency on one side of the parallelogram is 3. Find the area of the parallelogram.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{75}{2}}
Calculate the integral \( \int_{\gamma} \sqrt{z} \, dz \) where \( \gamma = C(2,1)^+ \), integrating over the principal value of \( \sqrt{z} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A cylinder is filled with gas at atmospheric pressure (103.3 kPa). Assuming the gas is ideal, determine the work (in joules) during the isothermal compression of the gas by a piston that has moved inside the cylinder by $h$ meters.

Hint: The equation of state for the gas is given by $\rho V=$ const, where $\rho$ is pressure and $V$ is volume.

Given:
$$
H=0.4 \text{ m}, \ h=0.2 \text{ m}, \ R=0.1 \text{ m}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{900}
A sphere is tangent to all the edges of the pyramid \( SABC \), specifically the lateral edges \( SA, SB, \) and \( SC \) at the points \( A', B' \), and \( C' \) respectively. Find the volume of the pyramid \( SA'B'C' \), given that \( AB = BC = SB = 5 \) and \( AC = 4 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2 \sqrt{59}}{15}}
There were electronic clocks on the International Space Station, displaying time in the format HH:MM. Due to an electromagnetic storm, the device started malfunctioning, and each digit on the display either increased by 1 or decreased by 1. What was the actual time when the storm occurred if the clocks showed 00:59 immediately after it?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11:48}
设 $\overrightarrow{O A}$ 与 $\overrightarrow{O B}$ 的夹角为 $\theta,|\overrightarrow{O A}|=2,|\overrightarrow{O B}|=1, \overrightarrow{O P}=t \overrightarrow{O A}, \overrightarrow{O Q}=$ $(1-t) \overrightarrow{O B}$, 函数 $f(t)=|\overrightarrow{P Q}|$ 在 $t=t_{0}$ 时取最小值. 若 $0<t_{0}<\frac{1}{5}$, 求 $\theta$ 的取值范围。答案的标准形式为$\left(\frac{\pi}{a}, \frac{b \pi}{c}\right)$，请给出区间端点a+b+c之和。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Remi prepared a tomato nursery and planted tomato seedlings. After 20 days, the seedlings were ready to be transferred. On the first day, he planted 200 seedlings on the farm. On the second day, while working alongside his father, he planted twice the number of seedlings he planted on the first day. On the third day, his father planted three times the amount Remi planted, and on the fourth day, his father planted four times the amount Remi planted. If the total number of seedlings transferred to the farm on these four days was 5000, how many seedlings did his father plant?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4400}
[ $\left[\begin{array}{l}\text { Auxiliary circle } \\ {[\text { Law of Sines }}\end{array}\right]$

In quadrilateral $A B C D$, it is known that $\angle A B D=\angle A C D=45^{\circ}, \angle B A C=30^{\circ}, B C=1$. Find $A D$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Example 10 Let $x, y, z$ be positive real numbers, and satisfy $x y z + x + z = y$, find
$$
p=\frac{2}{x^{2}+1}-\frac{2}{y^{2}+1}+\frac{3}{z^{2}+1}
$$

the maximum value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{10}{3}}
In a physical education class, 25 students from class 5"B" stood in a line. Each student is either an honor student who always tells the truth or a troublemaker who always lies.

Honor student Vlad took the 13th position. All the students, except Vlad, said: "There are exactly 6 troublemakers between me and Vlad." How many troublemakers are in the line in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Two real numbers  $x$  and  $y$  are such that  $8y^4+4x^2y^2+4xy^2+2x^3+2y^2+2x=x^2+1$ .
Find all possible values of  $x+2y^2$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
If the sides of a square are multiplied by 10, how many times as large is the area of the original square compared to the area of the resultant square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
For positive integer  $n$ , let  $s(n)$  denote the sum of the digits of  $n$ . Find the smallest positive integer  $n$  satisfying  $s(n)=s(n+864)=20$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{695}
Mia is helping her mom clean up 30 toys scattered on the floor. Every 30 seconds, her mom puts three toys into the storage box, and at the same time, Mia takes out two of the toys that were put in during the previous 30 seconds. After how many minutes will Mia and her mom first put all the toys into the storage box?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
The bases of trapezoid \(ABCD\) are related by the equation \(AD = 4 \cdot BC\). The sum of the angles \(\angle A + \angle D\) is \(120^\circ\). Points \(M\) and \(N\) are chosen on the legs of the trapezoid such that \(CN:ND = BM:MA = 1:2\). Perpendiculars drawn at points \(M\) and \(N\) to the legs of the trapezoid intersect at point \(O\). Find \(AD\), given that \(AO = 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
If a certain negative integer multiplied by a and -8b are negative integers, then the product of these integers plus ab is 89. What is the integer multiplied by a?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-11}
Let $ABCD$ be a square. Points $E$ and $F$ lie outside of $ABCD$ such that $ABE$ and $CBF$ are equilateral triangles. If $G$ is the centroid of triangle $DEF$, then find $\angle AGC$, in degrees
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$120^{\circ}$}
What is $\sqrt[2015]{2^01^5}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
1}
Given the sets $A=\left\{{x|{x=\frac{k}{6}+1,k∈Z}}\right\}$, $B=\left\{{x|{x=\frac{k}{3}+\frac{1}{2},k∈Z}}\right\}$, and $C=\left\{{x|{x=\frac{{2k}}{3}+\frac{1}{2},k∈Z}}\right\}$, determine the subset relationships among A, B, and C. Express your answer using ⊆ to denote subset relationships, and include all sets in your answer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{C \subseteq B \subseteq A}
What is the maximum number of non-attacking rooks that can be placed on an $n \times n \times n$ cube, where each rook can attack any square reachable vertically or horizontally, spanning a total of $4n$ squares in each direction?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^2}
Given the proposition ∀x > 1, x² + x + 1 > 0, what is its negation? Express your answer as a logical statement.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{∃x > 1, x² + x + 1 ≤ 0}
An elementary school teacher is making Halloween goodie bags for her class. She wants the bags to be personalized, so she surveys her students asking whether they'd like a vampire-themed bag or a pumpkin-themed bag. Of her 25 students, 11 indicate they want the vampire-themed bag and 14 indicate they want the pumpkin-themed bag. The store the teacher shops at sells packs of 5 of each theme at a price of $3 per package, as well as individual bags of each theme at a price of $1 each. The store has multiple special offers: "Buy 3 packages, Get 1 package Free", "Buy 5 individual bags, Get 1 Free", and a "10% discount on the entire purchase if you spend more than $10". What is the minimum amount of money the teacher can spend on the bags if she buys every student the theme they requested while maximizing the benefits of these special offers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$14.40}
For a set $A = \{1, 2, 3, \ldots, n\}$, determine the maximum size of a transitive relation $T$ on $A$ such that $T \neq A \times A$. What is the largest possible size for $T$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^2 - n + 1}
Will has a magic coin that can remember previous flips. If the coin has already turned up heads $m$ times and tails $n$ times, the probability that the next flip turns up heads is exactly \(\frac{m+1}{m+n+2}\). Suppose that the coin starts at $0$ flips. The probability that after $10$ coin flips, heads and tails have both turned up exactly $5$ times can be expressed as \(\frac{m}{n}\) for relatively prime positive integers $m$ and $n$. Compute $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}