dungnv/qwen36-27b-length-traces huggingface dataset:
user
The taxicab distance between points $\left(x_{1}, y_{1}, z_{1}\right)$ and $\left(x_{2}, y_{2}, z_{2}\right)$ is given by

$$
d\left(\left(x_{1}, y_{1}, z_{1}\right),\left(x_{2}, y_{2}, z_{2}\right)\right)=\left|x_{1}-x_{2}\right|+\left|y_{1}-y_{2}\right|+\left|z_{1}-z_{2}\right| .
$$

The region $\mathcal{R}$ is obtained by taking the cube $\{(x, y, z): 0 \leq x, y, z \leq 1\}$ and removing every point whose taxicab distance to any vertex of the cube is less than $\frac{3}{5}$. Compute the volume of $\mathcal{R}$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
## Real Expansion



Consider a "real" monatomic gas consisting of $N$ atoms of negligible volume and mass $m$ in equilibrium inside a closed cubical container of volume $V$. In this "real" gas, the attractive forces between atoms is small but not negligible. Because these atoms have negligible volume, you can assume that the atoms do not collide with each other for the entirety of the problem.
Context question:
a. Consider an atom in the interior of this container of volume $V$. Suppose the potential energy of the interaction is given by


$$

u(r)= \begin{cases}0 & r<d \\ -\epsilon\left(\frac{d}{r}\right)^{6} & r \geq d\end{cases}

$$



where $d \ll V^{1 / 3}$ is the minimum allowed distance between two atoms. Assume the gas is uniformly distributed within the container, what is the average potential energy of this atom? Write your answer in terms of $a^{\prime}=\frac{2 \pi d^{3} \epsilon}{3}, N$, and $V$.
Context answer:
\boxed{$-2 a^{\prime} N / V$}


Context question:
b. What is the average potential energy of an atom near the boundary of the box? Assume that there is no interaction between atoms near the boundary and the box itself.
Context answer:
\boxed{$-a^{\prime} N / V$}


Context question:
c. Using Bernoulli's law $P+U+\rho v^{2} / 2=$ constant, with pressure $P$, potential energy density $U$, mass density $\rho$ and fluid velocity $v$, what is the pressure at the boundary of the box? Assume the interior pressure is given by the ideal gas law.
Context answer:
\boxed{$\frac{N k T}{V}-a^{\prime} \frac{N^{2}}{V^{2}}$}


Context question:
d. Assuming most atoms are in the interior of the box, what is the total energy of the atoms in the box?
Context answer:
\boxed{$\frac{3}{2} N k T-a^{\prime} N^{2} / V$}


Extra Supplementary Reading Materials:

Now consider an insulated partitioned container with two sections, each of volume $V$. We fill one side of the container with $N$ atoms of this "real" gas at temperature $T$, which the other side being a vacuum. We then quickly remove the partition and let the gas expand to fill the entirety of the partitioned container. During this expansion, the energy of the gas remains unchanged.

e. What is the final temperature of the gas after the expansion?
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
An $\boldsymbol{n}$-label is a permutation of the numbers 1 through $n$. For example, $J=35214$ is a 5 -label and $K=132$ is a 3 -label. For a fixed positive integer $p$, where $p \leq n$, consider consecutive blocks of $p$ numbers in an $n$-label. For example, when $p=3$ and $L=263415$, the blocks are 263,634,341, and 415. We can associate to each of these blocks a $p$-label that corresponds to the relative order of the numbers in that block. For $L=263415$, we get the following:

$$
\underline{263} 415 \rightarrow 132 ; \quad 2 \underline{63415} \rightarrow 312 ; \quad 26 \underline{341} 5 \rightarrow 231 ; \quad 263 \underline{415} \rightarrow 213
$$

Moving from left to right in the $n$-label, there are $n-p+1$ such blocks, which means we obtain an $(n-p+1)$-tuple of $p$-labels. For $L=263415$, we get the 4 -tuple $(132,312,231,213)$. We will call this $(n-p+1)$-tuple the $\boldsymbol{p}$-signature of $L$ (or signature, if $p$ is clear from the context) and denote it by $S_{p}[L]$; the $p$-labels in the signature are called windows. For $L=263415$, the windows are $132,312,231$, and 213 , and we write

$$
S_{3}[263415]=(132,312,231,213)
$$

More generally, we will call any $(n-p+1)$-tuple of $p$-labels a $p$-signature, even if we do not know of an $n$-label to which it corresponds (and even if no such label exists). A signature that occurs for exactly one $n$-label is called unique, and a signature that doesn't occur for any $n$-labels is called impossible. A possible signature is one that occurs for at least one $n$-label.

In this power question, you will be asked to analyze some of the properties of labels and signatures.
Compute two other 6-labels with the same 4-signature as 462135.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Let $T=12$. Each interior angle of a regular $T$-gon has measure $d^{\circ}$. Compute $d$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Let $T=8$. At a party, everyone shakes hands with everyone else exactly once, except Ed, who leaves early. A grand total of $20 T$ handshakes take place. Compute the number of people at the party who shook hands with Ed.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Compute $\left\lfloor 100000(1.002)^{10}\right\rfloor$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Let $T=362$ and let $K=\sqrt{T-1}$. Compute $\left|(K-20)(K+1)+19 K-K^{2}\right|$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
3. The circular restricted three-body problem

In general, there is no exact solution of the three-body problem, in which three masses move under their mutual gravitational attraction. However, it is possible to make some progress by adding some constraints to the motion.

Two-body problem

Let's start with the motion of two masses, $M_{1}$ and $M_{2}$. Assume both masses move in circular orbits about their center of mass. Consider the inertial frame whose origin coincides with the center of mass of the system.
Context question:
(a) Express the equations of motion of $M_{1}$ and $M_{2}$ in terms of the gravitational constant $G$ and the position vectors $\overrightarrow{r_{1}}$ and $\overrightarrow{r_{2}}$ which point from the origin to $M_{1}$ and $M_{2}$, respectively.
Context answer:
$$
\begin{aligned}
& \frac{d^{2}}{d t^{2}} \overrightarrow{r_{1}}=\frac{G M_{2}}{\left|\overrightarrow{r_{2}}-\overrightarrow{r_{1}}\right|^{3}}\left(\overrightarrow{r_{2}}-\overrightarrow{r_{1}}\right) \\
& \frac{d^{2}}{d t^{2}} \overrightarrow{r_{2}}=\frac{G M_{1}}{\left|\overrightarrow{r_{1}}-\overrightarrow{r_{2}}\right|^{3}}\left(\overrightarrow{r_{1}}-\overrightarrow{r_{2}}\right)
\end{aligned}
$$


Context question:
(b) Find the period $T$ and angular frequency $\omega$ of the orbital motion.
Context answer:
\boxed{$\omega=\sqrt{\frac{G\left(M_{1}+M_{2}\right)}{\left|\overrightarrow{r_{2}}-\overrightarrow{r_{1}}\right|^{3}}}$ , $T=2 \pi \sqrt{\frac{\left|\overrightarrow{r_{2}}-\overrightarrow{r_{1}}\right|^{3}}{G\left(M_{1}+M_{2}\right)}}$}


Extra Supplementary Reading Materials:

Circular restricted three-body problem

Let us transform to a non-inertial frame rotating with angular velocity $\vec{\omega}=(0,0, \omega)$ about an axis normal to the orbital plane of masses $M_{1}$ and $M_{2}$, with the origin at their center of mass. In this frame, $M_{1}$ and $M_{2}$ are stationary at the Cartesian coordinates $(-\alpha R, 0,0)$ and $((1-\alpha) R, 0,0)$ respectively. The third mass $m$ is not stationary in this frame; in this non-inertial frame its position is $\vec{r}(t)=(x(t), y(t), 0)$.

The masses satisfy $M_{1}, M_{2} \gg m$. Consider $m$ to be so small that it does not affect the motion of $M_{1}$ or $M_{2}$.
Context question:
(a) Express $\alpha$ in terms of $M_{1}$ and $M_{2}$.
Context answer:
\boxed{$\alpha=\frac{M_{2}}{M_{1}+M_{2}}$}


Context question:
(b) Let $\rho_{1}(t)$ and $\rho_{2}(t)$ be the distances from $m$ to $M_{1}$ and $M_{2}$ respectively. Express $\rho_{1}(t)$ and $\rho_{2}(t)$ in terms of the coordinates and constants given.
Context answer:
\boxed{$\rho_{1}(t)=\sqrt{(x(t)+\alpha R)^{2}+(y(t))^{2}}$ , $\rho_{2}(t)=\sqrt{(x(t)-(1-\alpha) R)^{2}+(y(t))^{2}}$}

(c) By considering the centrifugal acceleration $\omega^{2} \vec{r}$ and Coriolis acceleration $-2 \omega \times$ $\vec{v}$, find the acceleration $\frac{d^{2}}{d t^{2}} \vec{r}$ of the third mass in terms of the coordinates and constants given, including $\rho_{1}$ and $\rho_{2}$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Let $T=$ 4. Pyramid $L E O J S$ is a right square pyramid with base $E O J S$, whose area is $T$. Given that $L E=5 \sqrt{2}$, compute $[L E O]$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Polygon $A_{1} A_{2} \ldots A_{n}$ is a regular $n$-gon. For some integer $k<n$, quadrilateral $A_{1} A_{2} A_{k} A_{k+1}$ is a rectangle of area 6 . If the area of $A_{1} A_{2} \ldots A_{n}$ is 60 , compute $n$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Scalene triangle $A B C$ has perimeter 2019 and integer side lengths. The angle bisector from $C$ meets $\overline{A B}$ at $D$ such that $A D=229$. Given that $A C$ and $A D$ are relatively prime, compute $B C$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Compute the number of integers $n$ for which $2^{4}<8^{n}<16^{32}$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Let $T=25$. Suppose that $T$ people are standing in a line, including three people named Charlie, Chris, and Abby. If the people are assigned their positions in line at random, compute the probability that Charlie is standing next to at least one of Chris or Abby.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
In a short-track speed skating event, there are five finalists including two Canadians. The first three skaters to finish the race win a medal. If all finalists have the same chance of finishing in any position, what is the probability that neither Canadian wins a medal?
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
3. The circular restricted three-body problem

In general, there is no exact solution of the three-body problem, in which three masses move under their mutual gravitational attraction. However, it is possible to make some progress by adding some constraints to the motion.

Two-body problem

Let's start with the motion of two masses, $M_{1}$ and $M_{2}$. Assume both masses move in circular orbits about their center of mass. Consider the inertial frame whose origin coincides with the center of mass of the system.
Context question:
(a) Express the equations of motion of $M_{1}$ and $M_{2}$ in terms of the gravitational constant $G$ and the position vectors $\overrightarrow{r_{1}}$ and $\overrightarrow{r_{2}}$ which point from the origin to $M_{1}$ and $M_{2}$, respectively.
Context answer:
$$
\begin{aligned}
& \frac{d^{2}}{d t^{2}} \overrightarrow{r_{1}}=\frac{G M_{2}}{\left|\overrightarrow{r_{2}}-\overrightarrow{r_{1}}\right|^{3}}\left(\overrightarrow{r_{2}}-\overrightarrow{r_{1}}\right) \\
& \frac{d^{2}}{d t^{2}} \overrightarrow{r_{2}}=\frac{G M_{1}}{\left|\overrightarrow{r_{1}}-\overrightarrow{r_{2}}\right|^{3}}\left(\overrightarrow{r_{1}}-\overrightarrow{r_{2}}\right)
\end{aligned}
$$


Context question:
(b) Find the period $T$ and angular frequency $\omega$ of the orbital motion.
Context answer:
\boxed{$\omega=\sqrt{\frac{G\left(M_{1}+M_{2}\right)}{\left|\overrightarrow{r_{2}}-\overrightarrow{r_{1}}\right|^{3}}}$ , $T=2 \pi \sqrt{\frac{\left|\overrightarrow{r_{2}}-\overrightarrow{r_{1}}\right|^{3}}{G\left(M_{1}+M_{2}\right)}}$}


Extra Supplementary Reading Materials:

Circular restricted three-body problem

Let us transform to a non-inertial frame rotating with angular velocity $\vec{\omega}=(0,0, \omega)$ about an axis normal to the orbital plane of masses $M_{1}$ and $M_{2}$, with the origin at their center of mass. In this frame, $M_{1}$ and $M_{2}$ are stationary at the Cartesian coordinates $(-\alpha R, 0,0)$ and $((1-\alpha) R, 0,0)$ respectively. The third mass $m$ is not stationary in this frame; in this non-inertial frame its position is $\vec{r}(t)=(x(t), y(t), 0)$.

The masses satisfy $M_{1}, M_{2} \gg m$. Consider $m$ to be so small that it does not affect the motion of $M_{1}$ or $M_{2}$.

(a) Express $\alpha$ in terms of $M_{1}$ and $M_{2}$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Let $T=1$. In parallelogram $A B C D, \frac{A B}{B C}=T$. Given that $M$ is the midpoint of $\overline{A B}$ and $P$ and $Q$ are the trisection points of $\overline{C D}$, compute $\frac{[A B C D]}{[M P Q]}$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Let $T=3$. Regular hexagon $S U P E R B$ has side length $\sqrt{T}$. Compute the value of $B E \cdot S U \cdot R E$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Let $T=23$ . Compute the units digit of $T^{2023}+T^{20}-T^{23}$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
If $\log _{10} x=3+\log _{10} y$, what is the value of $\frac{x}{y}$ ?
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Let $n \geq 1$ be an integer. What is the maximum number of disjoint pairs of elements of the set $\{1,2, \ldots, n\}$ such that the sums of the different pairs are different integers not exceeding $n$ ?
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Compute $\cos ^{3} \frac{2 \pi}{7}+\cos ^{3} \frac{4 \pi}{7}+\cos ^{3} \frac{8 \pi}{7}$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Determine the smallest number $M$ such that the inequality

$$
\left|a b\left(a^{2}-b^{2}\right)+b c\left(b^{2}-c^{2}\right)+c a\left(c^{2}-a^{2}\right)\right| \leq M\left(a^{2}+b^{2}+c^{2}\right)^{2}
$$

holds for all real numbers $a, b, c$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
If $3\left(8^{x}\right)+5\left(8^{x}\right)=2^{61}$, what is the value of the real number $x$ ?
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
A bag contains 20 lavender marbles, 12 emerald marbles, and some number of orange marbles. If the probability of drawing an orange marble in one try is $\frac{1}{y}$, compute the sum of all possible integer values of $y$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Let $T=2$. Compute the product of $-T-i$ and $i-T$, where $i=\sqrt{-1}$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
A fair coin is flipped $n$ times. Compute the smallest positive integer $n$ for which the probability that the coin has the same result every time is less than $10 \%$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Suppose that $x$ and $y$ are real numbers with $3 x+4 y=10$. Determine the minimum possible value of $x^{2}+16 y^{2}$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
In triangle $A B C, A B=5, A C=6$, and $\tan \angle B A C=-\frac{4}{3}$. Compute the area of $\triangle A B C$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Quadrilateral $A R M L$ is a kite with $A R=R M=5, A M=8$, and $R L=11$. Compute $A L$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Suppose that $f(a)=2 a^{2}-3 a+1$ for all real numbers $a$ and $g(b)=\log _{\frac{1}{2}} b$ for all $b>0$. Determine all $\theta$ with $0 \leq \theta \leq 2 \pi$ for which $f(g(\sin \theta))=0$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
## Spin Cycle



Cosmonaut Carla is preparing for the Intergalactic 5000 race. She practices for her race on her handy race track of radius $R$, carrying a stopwatch with her. Her racecar maintains a constant speed $v$ during her practices. For this problem, you can assume that $v>0.1 c$, where $c$ is the speed of light.
Context question:
a. How much time elapses on Carla's stopwatch with each revolution?
Context answer:
\boxed{$\frac{2 \pi R \sqrt{1-v^{2} / c^{2}}}{v}$}


Extra Supplementary Reading Materials:

Carla decides to do a fun experiment during her training. She places two stationary clocks down: Clock A at the center of the race track, i.e. the origin; and Clock B at a point on the race track denoted as $(R, 0)$. She then begins her training.



For parts (b) through (d), we define Carla's inertial reference frame (CIRF) as an inertial reference frame in which Carla is momentarily at rest, and which has the same origin of coordinates as the lab frame. Thus, CIRF is a new inertial frame each moment. The times on the clocks and stopwatch are all calibrated such that they all read 0 in CIRF when she passes by Clock $B$ for the first time.
Context question:
b. In the lab frame (the reference frame of the clocks, which are at rest), what is the offset between Clock $A$ and Clock $B$ ?
Context answer:
\boxed{0}


Context question:
c. If Carla's stopwatch measures an elapsed time $\tau$, what does Clock A measure in CIRF?
Context answer:
\boxed{$t_{A}(\tau)=\frac{\tau}{\sqrt{1-v^{2} / c^{2}}}$}

d. If Carla's stopwatch measures an elapsed time $\tau$, what does Clock B measure in CIRF?
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Triangle $A B C$ has vertices $A(0,5), B(3,0)$ and $C(8,3)$. Determine the measure of $\angle A C B$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Determine all pairs $(a, b)$ of positive integers for which $a^{3}+2 a b=2013$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Points $A(k, 3), B(3,1)$ and $C(6, k)$ form an isosceles triangle. If $\angle A B C=\angle A C B$, determine all possible values of $k$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Determine all real numbers $a, b$ and $c$ for which the graph of the function $y=\log _{a}(x+b)+c$ passes through the points $P(3,5), Q(5,4)$ and $R(11,3)$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Let $T=2018$, and append the digits of $T$ to $\underline{A} \underline{A} \underline{B}$ (for example, if $T=17$, then the result would be $\underline{1} \underline{\underline{A}} \underline{A} \underline{B}$ ). If the resulting number is divisible by 11 , compute the largest possible value of $A+B$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
## Spin Cycle



Cosmonaut Carla is preparing for the Intergalactic 5000 race. She practices for her race on her handy race track of radius $R$, carrying a stopwatch with her. Her racecar maintains a constant speed $v$ during her practices. For this problem, you can assume that $v>0.1 c$, where $c$ is the speed of light.

a. How much time elapses on Carla's stopwatch with each revolution?
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
What is the sum of the digits of the integer equal to $\left(10^{3}+1\right)^{2}$ ?
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Determine all values of $x$ for which $0<\frac{x^{2}-11}{x+1}<7$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Let $A B C$ be a triangle with $A B=A C$. The angle bisectors of $A$ and $B$ meet the sides $B C$ and $A C$ in $D$ and $E$, respectively. Let $K$ be the incenter of triangle $A D C$. Suppose that $\angle B E K=45^{\circ}$. Find all possible values of $\angle B A C$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Square $K E N T$ has side length 20 . Point $M$ lies in the interior of $K E N T$ such that $\triangle M E N$ is equilateral. Given that $K M^{2}=a-b \sqrt{3}$, where $a$ and $b$ are integers, compute $b$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Let T be an integer. Given a nonzero integer $n$, let $f(n)$ denote the sum of all numbers of the form $i^{d}$, where $i=\sqrt{-1}$, and $d$ is a divisor (positive or negative) of $n$. Compute $f(2 T+1)$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Determine all values of $x$ for which $(\sqrt{x})^{\log _{10} x}=100$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Bobby, Peter, Greg, Cindy, Jan, and Marcia line up for ice cream. In an acceptable lineup, Greg is ahead of Peter, Peter is ahead of Bobby, Marcia is ahead of Jan, and Jan is ahead of Cindy. For example, the lineup with Greg in front, followed by Peter, Marcia, Jan, Cindy, and Bobby, in that order, is an acceptable lineup. Compute the number of acceptable lineups.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Determine all real values of $x$ for which

$$
3^{x+2}+2^{x+2}+2^{x}=2^{x+5}+3^{x}
$$
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Define $f(x)=\sin ^{6} x+\cos ^{6} x+k\left(\sin ^{4} x+\cos ^{4} x\right)$ for some real number $k$.
Determine all real numbers $k$ for which $f(x)$ is constant for all values of $x$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Compute $\sin ^{2} 4^{\circ}+\sin ^{2} 8^{\circ}+\sin ^{2} 12^{\circ}+\cdots+\sin ^{2} 176^{\circ}$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Let $T=17$. In ARMLovia, the unit of currency is the edwah. Janet's wallet contains bills in denominations of 20 and 80 edwahs. If the bills are worth an average of $2 T$ edwahs each, compute the smallest possible value of the bills in Janet's wallet.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
On the complex plane, the parallelogram formed by the points $0, z, \frac{1}{z}$, and $z+\frac{1}{z}$ has area $\frac{35}{37}$, and the real part of $z$ is positive. If $d$ is the smallest possible value of $\left|z+\frac{1}{z}\right|$, compute $d^{2}$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
The perimeter of equilateral $\triangle P Q R$ is 12. The perimeter of regular hexagon $S T U V W X$ is also 12. What is the ratio of the area of $\triangle P Q R$ to the area of $S T U V W X$ ?
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Determine all pairs $(f, g)$ of functions from the set of positive integers to itself that satisfy

$$
f^{g(n)+1}(n)+g^{f(n)}(n)=f(n+1)-g(n+1)+1
$$

for every positive integer $n$. Here, $f^{k}(n)$ means $\underbrace{f(f(\ldots f}_{k}(n) \ldots))$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>
user
Given quadrilateral $A R M L$ with $A R=20, R M=23, M L=25$, and $A M=32$, compute the number of different integers that could be the perimeter of $A R M L$.
Please reason step by step, and put your final answer within \boxed{}.


assistant
<think>