inclusionAI/Ring-lite-rl-data huggingface dataset:
Consider a Markov chain $Z_k$ with 6 states, depicted as a pyramid where each node has a probability of $1/2$ to move to a neighboring node. Starting from state $1$ with $P(Z_0=1)=1$, determine the expected time $E[T_B]$ to first reach the bottom states, which are $3$, $4$, or $5$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In triangle \( ABC \), it is known that \(\angle BAC = 75^\circ\), \( AB = 1 \), and \( AC = \sqrt{6} \). On the side \( BC \), a point \( M \) is chosen such that \(\angle BAM = 30^\circ\). The line \( AM \) intersects the circumcircle of triangle \( ABC \) at a point \( N \) different from \( A \). Find \( AN \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the least positive integer $n$ such that the polynomial $P(X) = \sqrt{3} \cdot X^{n+1} - X^n - 1$ has at least one root of modulus $1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
[ Motion problems ]

A cyclist rode from point A to point B, where he stayed for 30 minutes, and then returned to A. On the way to B, he overtook a pedestrian, and 2 hours later met him on the return trip. The pedestrian arrived in B at the same time the cyclist returned to A. How much time did it take the pedestrian to travel from A to B, if his speed is four times less than the cyclist's speed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Given the function $$f(x) = \int_{B(0,R)} \log\left(\frac{\left\vert\,x - y\,\right\vert} {\left\vert\,x - \frac{R^2}{\left\vert\left\vert\,y\,\right\vert\right\vert^2}y\,\right\vert}\right)\,dy$$ for $x \in B(0,R)$, find the minimum value of $f(x)$ within the ball $B(0,R)$, which depends only on the radius $R$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\pi R^2}
Let $f$ be a polynomial. We say that a complex number $p$ is a double attractor if there exists a polynomial $h(x)$ so that $f(x)-f(p) = h(x)(x-p)^2$ for all x \in R. Now, consider the polynomial $$f(x) = 12x^5 - 15x^4 - 40x^3 + 540x^2 - 2160x + 1,$$ and suppose that it’s double attractors are $a_1, a_2, ... , a_n$. If the sum $\sum^{n}_{i=1}|a_i|$ can be written as $\sqrt{a} +\sqrt{b}$, where $a, b$ are positive integers, find $a + b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49}
$f(u)=u^{2}+a u+b-2, u=x+\frac{1}{x}, f(u)$ 有零点, 则 $a^{2}+b^{2}$ 的最小值为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
9 |  |

In a square table of $9 \times 9$ cells, 9 cells are marked at the intersections of the second, fifth, and eighth rows with the second, fifth, and eighth columns. How many ways are there to get from the bottom-left cell to the top-right cell, moving only through unmarked cells upwards or to the right?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{678}
Let $Z_1,Z_2,Z_3,Z_4 \in\mathbb{C}$ such that $Z_1+Z_2+Z_3+Z_4=0$ and $|Z_1|^2+|Z_2|^2+|Z_3|^2+|Z_4|^2=1$. What is the minimum value of $|Z_1-Z_2|^2+|Z_2-Z_3|^2+|Z_3-Z_4|^2+|Z_4-Z_1|^2$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
14. Teacher $D$ placed cards with the numbers $1, 2, \cdots, 13$ in a clockwise order on a circle. Initially, all cards are face up. Each time, one card can be flipped to face down (once a card is face down, it cannot be flipped again). Teacher $D$'s rule for flipping cards is: if a face-up card has the number $A$, and the card that is 2 positions away is also face up, then Teacher $D$ can flip the card with the number $A$. For example, as long as the cards with the numbers 9 or 2 are face up, Teacher $D$ can flip the card with the number 12 (of course, provided that the card with the number 12 is still face up). In the end, if Teacher $D$ can flip 12 cards to face down, then Teacher $D$ is successful. To achieve success, how many different sequences of flips can Teacher $D$ use?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26624}
The lateral face of a regular quadrilateral pyramid forms a $45^{\circ}$ angle with the base plane. Find the angle between the adjacent lateral faces.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120^\circ}
Calculate the areas of the figures bounded by the lines given in polar coordinates.

$$
r = 4 \cos 4 \phi
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8\pi}
In a neighborhood having 90 households, 11 did not have either a car or a bike. If some households had both a car and a bike and 44 had a car, there were 35 households with a bike only. How many households had both a car and a bike?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Evaluate the integral:

$$\int_{\tan x}^{\cot x} \cfrac{tdt}{1+t^2} + \int_{\cot x}^{\tan x} \cfrac{dt}{1+t^2}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
3. (17 points) In a $5 \times 5$ grid, 6 crosses need to be placed such that each row and each column contains at least one cross. How many ways can this be done?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4200}
What is the depth of the quotient ring $R/(p_1\cap p_2)$, where $R=K[x_1,\dots,x_n]$ is a polynomial ring over a field $K$, $I$ is a squarefree monomial ideal in $R$, and $p_1, p_2$ are distinct minimal prime ideals of $I$, each generated by subsets of the variables $\{x_1,\dots,x_n\}$ such that $I = p_1 \cap p_2$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given positive numbers $a$ and $b$ satisfying $\ln \frac{{2-b}}{a}=2a+2b-4$, find the minimum value of $\frac{1}{a}+\frac{2}{b}+\frac{2}{{ab}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5+2\sqrt{6}}{2}}
Let  $x$ - minimal root of equation  $x^2-4x+2=0$ . 
Find two first digits of number  $ \{x+x^2+....+x^{20} \}$  after  $0$ , where  $\{a\}$ - fractional part of  $a$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41}
The founder of a noble family received a plot of land. Each man in the family, upon his death, divided the land he had equally among his sons. If he had no sons, the land was transferred to the state. No other family members acquired or lost land by any other means. In total, there were 150 people in the family. What is the smallest possible fraction of the original plot that could be received by a family member?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2 \cdot 3^{49}}}
What is the tenth number in the row of Pascal's triangle that has 100 numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\binom{99}{9}}
Given the complex polynomial \( p(z) = z^4 + z^3 + 4z^2 + 3z + 7 \), determine the change in the argument along the curve \( C_3 \) of a quarter circle of sufficiently large radius centered at the origin. Assume there are no real or imaginary zeros. Express the argument of \( p(iy) \) as \( \text{arg}(p(iy)) = \tan^{-1} \left\{ \frac{y(\sqrt{3} - y)(\sqrt{3} + y)}{(y^2 - 2 - \sqrt{3}i)(y^2 - 2 + \sqrt{3}i)} \right\} \). Calculate the change in this argument along \( C_3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Find the number of solutions in positive integers to the equation \(x_1 + x_2 + x_3 + x_4 = 19\) subject to the conditions \(x_2 \neq 2x_3\) and \(x_1 \neq x_2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{702}
Express the conjugate of the function \( g(x) = \inf_z f(x,z) \) in terms of the conjugate \( f^*(y,s) \) of \( f(x,z) \), where \( f(x,z) \) is convex in \((x,z)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{g^*(y) = f^*(y, 0)}
The author of this question was born on April 24, 1977. What day of the week was that?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Sunday
}
Find the bounded solutions to the differential equation: $$\alpha'''(x) =\alpha'(x) x^2 + \alpha(x) x.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the sum of the coefficients in the expansion of \((x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7)^{11}\) where each variable has a degree of at least one.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{322494480}
Determine the number of bases required to test in order to conclusively determine if any integer less than \(2^{63} - 1\) is prime using the Miller-Rabin test.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Let $U_{\theta} \sim U[0,\theta]$ be independent for each $\theta \in [0,1]$. Define $X = \sup_{\theta \in [0,1]} U_{\theta}$. Find $\mathbb{P}(X \leq x)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x e^{1 - x}}
Find the equation of the curve that passes through the point (0,0) for the differential equation: \[ y'=\frac{\sin x + e^x (\sin x + \cos x) + x\cos x - y(1 + \cos x + e^x)}{e^x + \sin x + x - 2y} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y = \sin x}
Given integers \(a, b > 5\) such that \(\operatorname{lcm}(a,b)=8160\) and \(\gcd(a,b)=5\), find the pair \((a',b')\) where \(a=da', b=db'\) and \((a',b')=1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(32, 51)}
Determine the minimum number of weighings required to order 5 coins of different weights using a balance scale.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
A miner is trapped in a mine with three identical doors. The first door leads to safety in 2 hours. The second door leads back to the starting point in 3 hours. The third door leads to a maze, returning to the starting point in 5 hours on the first entry, and 1 hour on subsequent entries. The miner chooses each door with equal probability, independent of previous choices. What is the expected time until the miner reaches safety?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Evaluate the integral \( \frac{1}{2 \pi} \int_{0}^{2 \pi} \ln (|r e^{it} - z|) \, dt \) for \( r > |z| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln r}
Consider a linear transformation $x' = U x$ that preserves the inner product of vectors. If the transformation matrix $U$ has a determinant of $-1$, how many free parameters does $U$ have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n(n-1)}{2}}
Determine the largest value of $n$ as a function of $k$ such that a good order always exists, where a good order is defined as one in which each person either sees or hears both of their friends.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2k}
Determine the omega limit set of a trajectory in the following three-dimensional dynamical system:

$$\dot x = -y$$

$$\dot y = x$$

$$\dot z = -x^2-y^2$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
A cylinder has a fixed diameter of 50 mm and a length of 500 mm. Spherical balls have a diameter of 49.5 mm with a standard deviation of 0.1 mm. Estimate the probability that 12 of these balls will fit into the cylinder, considering both the width and length dimensions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given two independent random variables $X$ and $Y$ with the following distributions: $F_X(x) = \frac{1}{2}x$ for $0 < x < 2$ and $f_Y(y) = \frac{2}{9}y$ for $0 < y < 3$, find the probability that the product $XY \ge 1$. Provide your answer as a probability value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{25}{36}}
In the quadrilateral $E-ABCD$, the base $ABCD$ is a square, and $AE$ is perpendicular to the plane $CDE$. Given that $AE = DE = 3$, and $F$ is a point on the line segment $DE$, the dihedral angle $E-BC-F$ is equal to the dihedral angle $F-BC-D$. Find the length of $DF$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6\sqrt{5} - 12}
Calculate the probability that exactly 2 out of 4 couples sit next to each other when seated randomly in a row of 8 seats. Assume each couple consists of two people, and the seating arrangement is completely random.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{5}}
Determine the minimum number of points in a set $S$ in the plane such that for any point $X$ in the plane, there exists a point $M$ in $S$ where the distance $|XM|$ is rational.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the derivative of the expression $(Wx)^T A (Wx)$ with respect to the matrix $W$, where $W$ is an $m \times n$ matrix, $x$ is an $n \times 1$ vector, and $A$ is an $m \times m$ symmetric positive semi-definite matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 A W x x^T}
Given the initial and final positions of two stars orbiting a black hole, determine the coordinates of the black hole. Star 1 moves from (12, 2) to (10, 8), and Star 2 moves from (5, 6) to (2, 7). Find the coordinates of the black hole.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2, 2)}
Calculate the sum of the Fourier series for the 2π-periodic function \(f(x)=e^{x}\) at \(x=4\), given that the function is defined for \(|x|\le \pi\) and its Fourier series is \[f(x)=\frac{e^{\pi}-e^{-\pi}}{2\pi} + \frac{e^{\pi}}{\pi}\sum_{n=-\infty}^\infty \frac{(-1)^{n}}{1+n^{2}}e^{inx} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{4 - 2\pi}}
Find the smallest positive integer \( N \) such that the function \( f(x) = \sin \left(\frac{1}{x}\right) \) has exactly 2020 local extrema in the interval \( \left(\frac{1}{N}, 1\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6345}
A trough is 8 feet long and 1 foot high, with a vertical cross-section parallel to an end shaped like the graph of \( y = x^2 \) from \( x = -1 \) to \( x = 1 \). If the trough is full of water, calculate the amount of work required to pump all the water over the top of the trough.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{266.24}
Evaluate the determinant of the adjugate of an $n \times n$ matrix $A$, denoted as $\det(\text{adj}(A))$. Provide the correct expression for this determinant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(\det(A))^{n-1}}
Let $g:\mathbb{R} \to \mathbb{R}$ be a differentiable function such that $g'(x) < x^3$ for every $x > 0$. Determine the value of $$\lim_{x \to \infty} g(x).$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\infty}
Calculate the sum of the series \( \sum_{n=1}^\infty \frac{f(n)}{n(n+1)} \), where \( f(n) \) represents the number of 1's in the binary representation of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \ln 2}
Find the minimum value of the integral

$$ \mathcal {I} [y] = \int_{- \infty}^{\infty} \dfrac{1}{2} y'^2 + 1 - \cos y \; \mathrm d x $$

subject to the boundary conditions $y(-\infty) = 0$ and $y(\infty) = 2 \pi$. \( \boxed{\text{Minimum value}} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Find integers \((a, b)\) such that \(\gcd(a, b) = 13\) and they satisfy the equation \(49a - 125b = 169\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1131, 442)}
Determine the shortest sequence of letters using the alphabet A-E that guarantees the inclusion of any possible code of length up to 4. The code panel will only remember the last number of buttons pressed equal to the length of the code. For example, if the code is 2 letters long and you entered ABECD, the panel would only remember CD.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{628}
Find the limit distribution of \( Z_n = \sqrt{n} \cdot \min\{Y_1, Y_2, \ldots, Y_n \} \) as \( n \to \infty \), where \( Y_1, Y_2, \ldots \) are independent random variables uniformly distributed on \([0,1]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2-4x+1=0\), find the value of 
\[f(\alpha, \beta) =\frac {\beta^3}{2} \csc^2\left (\frac {1}{2}\arctan \frac {\beta}{\alpha}\right) +\frac {\alpha^3}{2} \sec^2\left (\frac {1}{2}\arctan \frac {\alpha}{\beta}\right)\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
In a vector space $R$ of dimension 6, let $A$, $B$, and $C$ be subspaces each of dimension 4. The dimension of the intersection of any two subspaces (i.e., $A \cap B$, $A \cap C$, and $B \cap C$) is 3. Determine the possible dimensions for the intersection of all three subspaces, $A \cap B \cap C$. Provide your answer as a set of possible dimensions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{2, 3\}}
Given a square matrix $B$ with spectral radius $\rho(B) > 1$, determine a matrix norm $||| \cdot |||$ such that $|||B||| > 1$ using the abstract method of defining an adapted norm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|||B||| > 1}
Let \( f(x) \) be a continuous function satisfying \( f(x^2+1) = \frac{x^2}{f(2^x)-1} \) and \( f(x) > 0 \) for all \( x \in \mathbb{R} \). Find \( \lim_{x\rightarrow 1}f(x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In how many ways can 9 distinct items be distributed into three boxes so that one box contains 3 items, another contains 2 items, and the third contains 4 items?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7560}
Determine the convergence rate of the iterative method defined by:
\[ x_{k+1} = x_k - \frac{f(x_k)}{g(x_k)} , \quad g(x_k) = \frac{f(x_k + f(x_k)) - f(x_k)}{f(x_k)} \]
where \( f(x) \in C^2[a,b] \) and there exists a point \( x_0 \) such that \( f(x_0) = 0 \) and \( f'(x_0) \neq 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the values of $k$ such that there exists a constant $c < 1$ for which the following inequality holds for all continuous functions $F$ defined on the interval $[0, r]$, where $r \in \mathbb{R}^+$:
\[ \sup_{x \in [0, r]} \left\{ e^{kx} \int_0^x |F(s)| \, ds \right\} \leq \frac{c}{M} \sup_{s \in [0, r]} \left\{ e^{ks} |F(s)| \right\} \]
where $M > 0$ is fixed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k < -M}
Determine the smallest natural number \( n \) such that in any two-coloring of the edges of the complete graph \( K_n \), there are always two monochromatic triangles that share exactly one common vertex.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Determine the holomorphic function \( f: \mathbb{C} \rightarrow \mathbb{C} \) such that for all \( x, y \in \mathbb{R} \), the imaginary part of \( f(x + iy) \) is equal to \( x^2 - y^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{i z^2}
Determine the minimum number of repetitions possible when superimposing two different Latin squares of the same order.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the limit distribution of the sequence of random variables $X_1, X_2, \ldots$, where $P(X_n = \frac{k}{n}) = \frac{1}{n}$ for $k = 1, 2, \ldots, n$, as $n$ approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{U(0,1)}
Evaluate the integral \( \int\limits_0^{\pi}\log (1-\cos x)\, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\pi \ln 2}
Determine whether the sequence \( a_n = \sqrt[n]{|\cos n|} \) converges or diverges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
If $f(x)$ is a function such that $|f(x)|\le 1, x \in R$, and $$g(x)=\frac{e^{f(x)}-e^{|f(x)|}}{e^{f(x)}+e^{|f(x)|}},$$ then what is the maximum value of $g(x)$ if it is guaranteed that $f(x)$ covers the entire range $[-1,1]$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
If 5% more is gained by selling an article for a higher price than by selling it for Rs. 340, the cost of the article is Rs. 400. What is the higher selling price?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{360}
a is twice as good a workman as b, and c is thrice as good as a. If they took 9 days together to do the work, and b worked 2 days less than a and c, how many days will b alone take to complete the work?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{81}
A $5 \times 5$ grid of squares is filled with integers. Call a rectangle corner-odd if its sides are grid lines and the sum of the integers in its four corners is an odd number. What is the maximum possible number of corner-odd rectangles within the grid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Find all positive integers  $k$  such that there exists a positive integer  $n$ , for which  $2^n + 11$  is divisible by  $2^k - 1$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ k = 4 }
The number of significant digits in the measurement of the side of a square whose computed area is $1.1025$ square inches to the nearest ten-thousandth of a square inch is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Determine the coefficient of $x^4$ in the Taylor series expansion of $\frac{\tan x}{x}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{15}}
Determine all real numbers \(x\) for which the series \(\sum_{n=2}^{\infty} \frac{\sin(nx)}{\log n}\) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}}
Let $S(t)$, $t \geq 0$ be a geometric Brownian motion with drift parameter $\mu = 0.2$ and volatility parameter $\sigma = 0.2$. Calculate the probability $P(S(3) > S(1))$. Provide your answer as a single probability value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.8985}
Find the volume of the union $A \cup B \cup C$ of the three subsets $A$, $B$, $C$ in $xyz$ space such that:

$$
 A = \{(x, y, z) \mid |x| \leq 1, \, y^2 + z^2 \leq 1\} 
$$

$$
 B = \{(x, y, z) \mid |y| \leq 1, \, z^2 + x^2 \leq 1\} 
$$

$$
 C = \{(x, y, z) \mid |z| \leq 1, \, x^2 + y^2 \leq 1\} 
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2(3\pi - 4\sqrt{2})$
}
Let $X_1, X_2, X_3$ be independent random variables, each following a normal distribution with mean $0$ and variance $\sigma^2$. Determine the distribution of the random variable:

$$Y=\frac{1}{3}\left(\left(\frac{X_1}{\sigma}-\frac{X_2}{\sigma}\right)^2+\left(\frac{X_2}{\sigma}-\frac{X_3}{\sigma}\right)^2+\left(\frac{X_3}{\sigma}-\frac{X_1}{\sigma}\right)^2\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\chi^2(2)}
Determine the maximal possible dimension of an abelian Lie subalgebra of the Heisenberg Lie algebra \( \mathfrak{h}_n \) of dimension \( 2n+1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n + 1}