inclusionAI/Ring-lite-rl-data huggingface dataset:
Given that a blue ball and an orange ball are randomly and independently tossed into bins numbered with the positive integers, where for each ball the probability that it is tossed into bin k is 3^(-k) for k = 1, 2, 3, ..., determine the probability that the blue ball is tossed into a higher-numbered bin than the orange ball.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7}{16}}
Given \(a > 0\), \(b > 0\), two lines \(l_1:(a-1)x+y-1=0\) and \(l_2:x+2by+1=0\), and \(l_1\) is perpendicular to \(l_2\), find the minimum value of \(\dfrac{2}{a}+\dfrac{1}{b}\). Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Find the surface area of the intersection of the paraboloid $x^2+y^2-z=0$ and the plane $z=2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2\pi\sqrt{2}$}
3 [ Constructions on a Projection Drawing ]

In a regular quadrilateral pyramid with a lateral edge of 20, the angle between the lateral edges lying in the same face is $\frac{\pi}{3}$. A line perpendicular to one of the lateral edges and passing through a point on this edge intersects the height of the pyramid. Find the length of the segment of this line that lies inside the pyramid, if the point of intersection of this line with the height divides the height in the ratio 3:7, counting from the vertex.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Let real numbers \( s \) and \( t \) satisfy \( 19s^2 + 99s + 1 = 0 \) and \( t^2 + 99t + 19 = 0 \), respectively, with \( st \neq 1 \). Find the value of \(\frac{st + 4s + 1}{t}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-5}
Evaluate the double integral $$\int_{0}^{1}\mathrm dy\!\int_{0}^{1}\frac{x-y}{(x+y)^3}\mathrm dx.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Let \( p = 2^{24036583} - 1 \), the largest prime currently known. For how many positive integers \( c \) do the quadratics \( \pm x^2 \pm p x \pm c \) all have rational roots?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Problem 2. On a plane, 100 points are marked. It turns out that on two different lines a and b, there are 40 marked points each. What is the maximum number of marked points that can lie on a line that does not coincide with a and b?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
$ABCDE$ is inscribed in a circle with $AB = BC = CD = DE = 4$ and $AE = 1.$  Compute $(1 - \cos \angle B)(1 - \cos \angle ACE).$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{64}}
If $\triangle A_{1} A_{2} A_{3}$ is an equilateral triangle and $A_{n+3}$ is the midpoint of segment $A_{n} A_{n+1}$ for all positive integers $n$, then find the degree measure of $\angle A_{44} A_{45} A_{43}$. Express your answer in degrees as a numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Square \(ABCD\) and square \(ABEF\) lie in planes that form an angle of \(120^\circ\). Points \(M\) and \(N\) are on diagonals \(AC\) and \(BF\), respectively, such that \(AM = FN\). If \(AB = 1\), determine the range of values for \(MN\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[ \frac{\sqrt{3}}{2}, 1 \right]}
How many rooks can be placed in an  $n\times n$  chessboard such that each rook is threatened by at most  $2k$  rooks?
(15 points)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ n(k+1) }
32nd CanMO 2000 Problem 2 How many permutations of 1901, 1902, 1903, ... , 2000 are such that none of the sums of the first n permuted numbers is divisible by 3 (for n = 1, 2, 3, ... , 2000)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{99!\cdot34!\cdot33!}{66!}}
On a table there were 2 green apples, 3 red apples, 14 yellow apples, 4 red oranges, 8 yellow oranges, 10 green kiwis, 7 purple grapes, and 5 green grapes. If someone wants to have a fruit platter with a ratio of 2 green apples to 4 red apples to 3 yellow apples, as well as one red orange, two yellow oranges, and an equal number of green kiwis and purple grapes, how many fruits will be in their platter?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
The measure of the angle $\angle A$ of the acute triangle $ABC$ is $60^o$, and $HI = HB$, where $I$ and $H$ are the incenter and the orthocenter of the triangle $ABC$. Find the measure of the angle  $\angle  B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80^\circ}
Let \(ABC\) be a triangle with \(AB=7\), \(BC=9\), and \(CA=4\). Let \(D\) be the point such that \(AB \parallel CD\) and \(CA \parallel BD\). Let \(R\) be a point within triangle \(BCD\). Lines \(\ell\) and \(m\) going through \(R\) are parallel to \(CA\) and \(AB\) respectively. Line \(\ell\) meets \(AB\) and \(BC\) at \(P\) and \(P'\) respectively, and \(m\) meets \(CA\) and \(BC\) at \(Q\) and \(Q'\) respectively. If \(S\) denotes the largest possible sum of the areas of triangles \(BPP'\), \(RP'Q'\), and \(CQQ'\), determine the value of \(S^2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{180}
Given an arithmetic sequence {a_n}, the sum of the first n terms is S_n, and it is known that S_8 = -3S_4!= 0. Find the value of S_4 / S_12. Express your answer as a fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1/12}
Cameron has $90$ red tokens and $60$ blue tokens. He can exchange three red tokens for one gold token and two blue tokens, or two blue tokens for one gold token and one red token. What is the total number of gold tokens Cameron ends up with after exchanging his tokens until no more exchanges can be made?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{148}
Marko lives at the origin of the Cartesian plane. Every second, Marko moves $1$ unit up with probability $\frac{2}{9}$, $1$ unit right with probability $\frac{2}{9}$, $1$ unit up and $1$ unit right with probability $\frac{4}{9}$, and he doesn’t move with probability $\frac{1}{9}$. After $2019$ seconds, Marko ends up at the point $(A, B)$. What is the expected value of $A \cdot B$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1811716}
若 $x$ 为实数, 使得 $2, x, x^{2}$ 互不相同，且其中有一个数恰为另一个数的 2倍，则这样的实数 $x$ 的个数为多少？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given \(\arcsin x < \arccos x < \operatorname{arccot} x\), the range of real number \(x\) is \(\quad\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(0, \frac{\sqrt{2}}{2}\right)}
Let $\triangle ABC$ be an equilateral triangle with side length $55.$ Points $D,$ $E,$ and $F$ lie on $\overline{BC},$ $\overline{CA},$ and $\overline{AB},$ respectively, with $BD = 7,$ $CE=30,$ and $AF=40.$ Point $P$ inside $\triangle ABC$ has the property that <cmath>\angle AEP = \angle BFP = \angle CDP.</cmath> Find $\tan^2(\angle AEP).$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{075}
The numbers from 1 to 9 are placed at the vertices of a cube such that the sum of the four numbers on each face is the same. Find the common sum.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22.5}
Find the greatest integer $n$ such that $10^n$ divides \[ \frac{2^{10^5} \cdot 5^{2^{10}}}{10^{5^2}} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{999}
How many four-digit positive integers have exactly one digit equal to $1$ and exactly one digit equal to $3$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{720}
A1. Professor Oak is feeding his 100 Pokémon. Each Pokémon has a bowl whose capacity is a positive real number of kilograms. These capacities are known to Professor Oak. The total capacity of all the bowls is 100 kilograms. Professor Oak distributes 100 kilograms of food in such a way that each Pokémon receives a non-negative integer number of kilograms of food (which may be larger than the capacity of their bowl). The dissatisfaction level of a Pokémon who received $N$ kilograms of food and whose bowl has a capacity of $C$ kilograms is equal to $|N-C|$.

Find the smallest real number $D$ such that, regardless of the capacities of the bowls, Professor Oak can distribute the food in a way that the sum of the dissatisfaction levels over all the 100 Pokémon is at most $D$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
How many different positions can appear on a chessboard if both players, starting from the initial position, make just one move each?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{400}
Given the equation of the hyperbola $\frac{x^2}{7} - \frac{y^2}{9} = 1$, find the coordinates of its foci. Express your answer as coordinates (x, y), including the ± symbol if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(±4, 0)}
After getting 3 successive discounts, a shirt with a list price of Rs 250 is available at Rs 150. If the second discount is 15%, and the third discount is 12%, find the first discount percentage.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19.79\%}
Let $(a_n)_{n\geq1}$ be a strictly increasing sequence of positive real numbers such that $\lim\limits_{n\to\infty} \frac{a_{n+1}}{a_n}=1$. Find
$$\lim_{n\to\infty} \sum_{k=1}^{n} \frac{a_k}{a_k+a_1+a_2+...+a_n}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Quadrilateral \(ABCD\) is inscribed in a circle. Diagonal \(AC\) is the bisector of angle \(BAD\) and intersects diagonal \(BD\) at point \(K\).

Find \(KC\), if \(BC = 4\) and \(AK = 6\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
# Problem 9.3

In a convex pentagon $P Q R S T$, angle $P R T$ is half the size of angle $Q R S$, and all sides are equal. Find angle $P R T$.

## Number of points 7
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
A roulette can land on any number from 0 to 2007 with equal probability. The roulette is spun repeatedly. Let $P_{k}$ be the probability that at some point the sum of the numbers that have appeared in all spins equals $k$. Which number is greater: $P_{2007}$ or $P_{2008}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{P_{2007}}
In a subject test, the average score of Xiaofang's four subjects: Chinese, Mathematics, English, and Science, is 88. The average score of the first two subjects is 93, and the average score of the last three subjects is 87. What is Xiaofang's English test score?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{95}
Half of Jerome's money was $43. He gave $8 to Meg and thrice as much to Bianca. After that, Jerome gave twice his remaining money to his friend Nathan. Finally, he decided to donate 20% of what he had left to charity. How much does Jerome have left now?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$21.60}
What is the minimum number of 3-cell L-shaped tiles that need to be painted in a $5 \times 5$ square so that no more L-shaped tiles can be painted? (The painted L-shaped tiles must not overlap.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A man can row a boat at 14 kmph in still water. If he rows downstream, where the speed of current is a certain speed, he takes 8.999280057595392 seconds to cover 40 meters. What is the speed of the current?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Dietitian A, B, and C each prepared a lunch consisting of two types of food, X and Y. Food X contains 5 calories per serving, and Food Y contains 8 calories per serving. They must maintain a ratio of 2 servings of Food X for every 3 servings of Food Y to meet their specific dietary requirements. 

Dietitian A ate three-fourths of her prepared lunch, Dietitian B ate five-sixths, and Dietitian C ate half of her meal during the lunch break. If the FDA recommends a maximum lunch intake of 25 calories for dietitians, how many more calories than the recommended amount was eaten collectively by the three dietitians, considering their specific ratio of servings for Food X and Food Y as well as the total calories in their lunch?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4.17}
Consider all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying $$f(f(x)+2 x+20)=15$$ Call an integer $n$ good if $f(n)$ can take any integer value. In other words, if we fix $n$, for any integer $m$, there exists a function $f$ such that $f(n)=m$. Find the sum of all good integers $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-35}
Given a geometric sequence {a_n}, the sum of its first n terms is denoted as S_n. If 4a_1, a_2, and a_3 form an arithmetic sequence, and a_1 = 1, calculate the sum of the first 4 terms, S_4. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
A rectangular garden with a perimeter of 480 m is divided into three sections: a large rectangle and two right-angled triangular sections on either side. The triangular sections have bases 50 m in length along the garden boundary. If the height of each triangular section is 100 m, what is the length of the rectangular section of the garden?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{140}
Given the functions f(x) = x^2 ln(x) and g(x) = x/e^x, find the range of the positive real number k such that there exist x1 in [e, e^2] and x2 in [1, 2] satisfying the inequality e^3(k^2-2)g(x2) >= kf(x1). Express your answer as an inequality involving k.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k\geq 2}
Triangle $ABC_0$ has a right angle at $C_0$. Its side lengths are pairwise relatively prime positive integers, and its perimeter is $p$. Let $C_1$ be the foot of the altitude to $\overline{AB}$, and for $n \geq 2$, let $C_n$ be the foot of the altitude to $\overline{C_{n-2}B}$ in $\triangle C_{n-2}C_{n-1}B$. The sum $\sum\limits_{n=2}^\infty C_{n-2}C_{n-1} = 6p$. Find $p$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{182}
Find the smallest positive integer $N$ such that $N$ is divisible by $19$, and when the digits of $N$ are read in reverse order, the result (after removing any leading zeroes) is divisible by $36$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2394}
A lateral edge of a regular triangular pyramid forms an angle of $45^{\circ}$ with the plane of the base. Find the angle between the apothem of the pyramid and the plane of an adjacent lateral face.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{arcsin \frac{3}{5}}
The Dinky is a train connecting Princeton to the outside world. It runs on an odd schedule: the train arrives once every one-hour block at some uniformly random time (once at a random time between $9\text{am}$ and $10\text{am}$, once at a random time between $10\text{am}$ and $11\text{am}$, and so on). One day, Emilia arrives at the station, at some uniformly random time, and does not know the time. She expects to wait for $y$ minutes for the next train to arrive. After waiting for an hour, a train has still not come. She now expects to wait for $z$ minutes. Find $yz$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{525}
In the diagram, \( B, C \) and \( D \) lie on a straight line, with \(\angle ACD=100^{\circ}\), \(\angle ADB=x^{\circ}\), \(\angle ABD=2x^{\circ}\), and \(\angle DAC=\angle BAC=y^{\circ}\). Find the value of \( x \). Express your answer in degrees as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Bobbo starts swimming at 2 feet per second across a 100-foot wide river with a current of 5 feet per second. Bobbo doesn't know that there is a waterfall 175 feet from where he entered the river. He realizes his predicament midway across the river. What is the minimum speed that Bobbo must increase to make it to the other side of the river safely?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 \text{ feet/second}}
Given the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$ with a point $P(1,-1)$ inside it and $F$ being the right focus of the ellipse, find the point $M$ on the ellipse such that the value of $|M P| + 2|M F|$ is minimized.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left( \frac{2\sqrt{6}}{3}, -1 \right)}
设复数 z 满足 |z|=1，且使得关于 x 的方程 zx^2+2\bar{z}x+2=0 有实根. 则这样的 z 的和为________.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{-1}
Let \(ABC\) be a triangle such that \(|AB| = 7\), \(|BC| = 8\), \(|AC| = 6\). Let \(D\) be the midpoint of side \([BC]\). If the circle through \(A\), \(B\), and \(D\) cuts \(AC\) at \(A\) and \(E\), find the length of \(|AE|\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Tim is participating in the following three math contests. On each contest, his score is the number of correct answers.
- The Local Area Inspirational Math Exam consists of 15 problems.
- The Further Away Regional Math League has 10 problems.
- The Distance-Optimized Math Open has 50 problems.

For every positive integer $n$, Tim knows the answer to the $n$th problems on each contest (which are pairwise distinct), if they exist; however, these answers have been randomly permuted so that he does not know which answer corresponds to which contest. Unaware of the shuffling, he competes with his modified answers. Compute the expected value of the sum of his scores on all three contests.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
In set operations, a set is in one-to-one correspondence with its complement in the universal set, forming a "set pair", which is often used in solving set problems. Now, suppose the universal set $U$ contains $11$ elements. For $k$ different subsets $A_{1}$, $A_{2}$, $\ldots$, $A_{k}$ of set $U$, where the intersection of any two subsets is not empty, and any other subset of $U$ has an empty intersection with at least one of $A_{1}$, $A_{2}$, $\ldots$, $A_{k}$, then $k=\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1024}
A 10x10 arrangement of alternating black and white squares has a black square $R$ in the second-bottom row and a white square $S$ in the top-most row. Given that a marker is initially placed at $R$ and can move to an immediately adjoining white square on the row above either to the left or right, and the path must consist of exactly 8 steps, calculate the number of valid paths from $R$ to $S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
The graph of the rational function $\frac{2x^6+3x^5 - x^2 - 1}{q(x)}$  has a horizontal asymptote. What is the smallest possible degree of $q(x)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let $x_n$ be a real sequence converging to $x$. What is the maximum number of limit points of the sequence $z_n=\lceil {x_n^3} \rceil-\lceil {x_n} \rceil$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Weight, $W(X)$: The smallest possible cardinality of a base is called the weight of the topological space. Let $X=\{a,b,c,d,e\}$ and $t=\{\emptyset,X,\{a\},\{b\},\{a,b\},\{a,b,c\}\}$ find the weight of $X$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $f(n) = 9n^5 - 5n^3 - 4n$. Find the greatest common divisor of $f(17), f(18), \ldots ,f(2009)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Given that Tian Ji's top horse is faster than the King of Qi's middle horse, and Tian Ji's middle horse is faster than the King of Qi's bottom horse, but Tian Ji's top horse is slower than the King of Qi's top horse, and Tian Ji's bottom horse is slower than the King of Qi's bottom horse, calculate the probability that the King of Qi's horse wins when one horse is randomly selected from each side for a race.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{3}}
Choose a permutation of $ \{1,2, ..., 20\}$  at random. Let  $m$  be the amount of numbers in the permutation larger than all numbers before it. Find the expected value of  $2^m$ .

*Proposed by Evan Chang (squareman), USA*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
In a positive term geometric sequence ${a_n}$, ${a_5 a_6 =81}$, calculate the value of ${\log_{3}{a_1} + \log_{3}{a_5} +...+\log_{3}{a_{10}}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
13. Given that $a$, $b$, $c$, are the lengths of the sides opposite to angles $A$, $B$, $C$ in $\triangle ABC$ respectively, with $a=2$, and $(2+b)(\sin A-\sin B)=(c-b)\sin C$, find the maximum area of $\triangle ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
Shaq sees the numbers $1$ through $2017$ written on a chalkboard. He repeatedly chooses three numbers, erases them, and writes one plus their median. For instance, if he erased $-2, -1, 0$, he would replace them with $0$. If $M$ is the maximum possible final value remaining on the board, and $m$ is the minimum, compute $M - m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2014}
Three equilateral triangles $ABC$, $BCD$, and $CDE$ are positioned such that $B$, $C$, and $D$ are collinear, and $C$ is the midpoint of $BD$. Triangle $CDE$ is positioned such that $E$ is on the same side of line $BD$ as $A$. What is the value of $AE \div BC$ when expressed in simplest radical form?

[asy]
draw((0,0)--(5,8.7)--(10,0)--cycle);
draw((10,0)--(12.5,4.35)--(15,0)--cycle);
label("$A$",(0,0),SW);
label("$B$",(5,8.7),N);
label("$C$",(10,0),S);
label("$D$",(15,0),SE);
label("$E$",(12.5,4.35),N);
[/asy]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
Estimate the range of the submissions for this problem. Your answer must be between  $[0, 1000]$ . An estimate  $E$  earns  $\frac{2}{1+0.05|A-E|}$  points, where  $A$  is the actual answer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{500}
B3. A digital clock displays times from 00:00:00 to 23:59:59 throughout a day. Every second of the day, you can add up the digits, resulting in an integer. At 13:07:14, for example, the sum is $1+3+0+7+1+4=16$. If you write down the sum for every possible time on the clock and then take the average of all these numbers, what is the result?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18\frac{3}{4}}
The height of a right prism is 1, and the base of the prism is a rhombus with a side length of 2 and an acute angle of $30^\circ$. A plane intersects the prism along a side of the base, and it is inclined to the base plane at an angle of $60^\circ$. Find the area of the cross-section.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{\sqrt{3}}}