inclusionAI/Ring-lite-rl-data huggingface dataset:
There are 4 points \( A, B, C, D \) in space, satisfying \( AB = BC = CD \). Given that \( \angle ABC = \angle BCD = \angle CDA = 36^\circ \), what is the angle between line \( AC \) and line \( BD \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
Given two positive numbers $a$ and $b$, a new number $c$ can be generated according to the rule $c=ab+a+b$. By selecting the two larger numbers among $a$, $b$, and $c$ and applying the aforementioned rule, a new number is generated. This process is repeated, and each generation of a new number is considered one operation. If $p > q > 0$, and after 6 operations, the generated number is $(q+1)^m(p+1)^n-1$ (where $m$ and $n$ are positive integers), then the value of $m+n$ is $\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
How many homomorphisms are there from the group $G = \mathbb{Z}/2 \times \mathbb{Z}/2$ to the symmetric group $S_3$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Let $p, q, r$ be distinct prime numbers and let 
\[ A = \{p^a q^b r^c \mid 0 \le a, b, c \le 5\} \]
Find the least $n \in \mathbb{N}$ such that for any $B \subset A$ where $|B| = n$, there exist elements $x$ and $y$ such that $x$ divides $y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
Given a convex n-gon, circles are drawn with each of its sides as diameters, ensuring that the n-gon can be completely covered by these n circles. Determine the maximum value of n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given a sequence of numbers arranged according to a certain rule: $-\frac{3}{2}$, $1$, $-\frac{7}{10}$, $\frac{9}{17}$, $\ldots$, based on the pattern of the first $4$ numbers, the $10$th number is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{21}{101}}
A hexagonal prism has a height of 165 cm. Its two hexagonal faces are regular hexagons with sides of length 30 cm. Its other six faces are rectangles. A fly and an ant start at point \(X\) on the bottom face and travel to point \(Y\) on the top face. The fly flies directly along the shortest route through the prism. The ant crawls around the outside of the prism along a path of constant slope so that it winds around the prism exactly \(n + \frac{1}{2}\) times, for some positive integer \(n\). The distance crawled by the ant is more than 20 times the distance flown by the fly. What is the smallest possible value of \(n\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
Calculate the limit: $\lim_{n \to \infty} \frac{P_n^2 + C_n^2}{(n+1)^2} = \_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{2}}
Let $X$ and $Y$ be independent Poisson random variables, both with parameter $\lambda$. Find a closed-form expression for $P(X \leq Y)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2} (1 + e^{-2\lambda} I_0(2\lambda))}
In a kilometer race, A beats B by some meters or 10 seconds. A takes 190 seconds to complete the race. By how many meters does A beat B?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
A box contains three balls, each of a different color. Every minute, Randall randomly draws a ball from the box, notes its color, and then returns it to the box. Consider the following two conditions: (1) Some ball has been drawn at least three times (not necessarily consecutively). (2) Every ball has been drawn at least once. What is the probability that condition (1) is met before condition (2)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{13}{27}}
In triangle $A B C$ with $A B=8$ and $A C=10$, the incenter $I$ is reflected across side $A B$ to point $X$ and across side $A C$ to point $Y$. Given that segment $X Y$ bisects $A I$, compute $B C^{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
In a school of witchcraft and wizardry, there are 13 students. Before the divination exam, the teacher seated them around a round table and asked them to guess who would receive a diploma in divination. Each student modestly refrained from naming themselves and their two neighbors, but wrote about everyone else: "None of these ten will receive it!" Of course, all those who passed the exam guessed correctly, and all the other students were wrong. How many wizards received the diploma?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the number of positive integers  $a \le 250$  for which the set  $\{a+1, a+2, \dots, a+1000\}$  contains $\bullet$  Exactly  $333$  multiples of  $3$ , $\bullet$  Exactly  $142$  multiples of  $7$ , and $\bullet$  Exactly  $91$  multiples of  $11$ .

*Based on a proposal by Rajiv Movva*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
What is the maximum number of planes of symmetry that a spatial figure consisting of three pairwise non-parallel lines can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
已知${{\left( \sqrt[3]{x}+\frac{2}{x} \right)}^{n}}$展开式中第二项、第三项、第四项的二项式系数成等差数列，将展开式中所有项重新排列，则有理项不相邻的概率为______．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{5}{14}$}
After Clive assembled and wound his clock (see problem $\underline{32798}$), setting it by his grandfather's, it started running backward. How many times a day will it show the correct time?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
If the 13th of any particular month falls on a Friday, we call it Friday the 13th. It is known that Friday the 13th occurs at least once every calendar year. If the longest interval between two consecutive occurrences of Friday the 13th is \( x \) months, find \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Evaluate the limit $$\lim_{x\rightarrow0}\frac{1-\cos\left(\frac{1-\cos x \cos 2x}{x^2}-\frac {5}{2}\right)\cos2x}{x^2}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In triangle \( ABC \), point \( M \) is the midpoint of \( AC \). Additionally, \( BC = \frac{2}{3}AC \) and \(\angle BMC = 2 \angle ABM\). Find the ratio \( AM / AB \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{2 \sqrt{5}}}
Given the expressions $a = \left(\frac{1}{3}\right)^{\frac{2}{3}}$, $b = \left(\frac{1}{5}\right)^{\frac{2}{3}}$, and $c = \left(\frac{4}{9}\right)^{\frac{1}{3}}$, determine the exact relationship between $a$, $b$, and $c$. Express your answer using less than (<) or greater than (>) symbols to indicate the relationships between the three expressions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{b < a < c}
There are five links of a chain, each with three rings.

What is the minimum number of rings that need to be broken and rejoined to connect these links into one continuous chain?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given $\triangle ABC \sim \triangle A'B'C'$, and the similarity ratio is $3:2$. If $A'B' = 10$, then $AB$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
In an equilateral triangle \(ABC\), points \(A_1\) and \(A_2\) are chosen on the side \(BC\) such that \(BA_1 = A_1A_2 = A_2C\). A point \(B_1\) is chosen on the side \(AC\) such that \(AB_1 : B_1C = 1 : 2\). Find the sum of the angles \(\angle AA_1 B_1\) and \(\angle AA_2 B_1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30^\circ}
A large number of rocks are placed on a table. On each turn, one may remove some rocks from the table following these rules: on the first turn, only one rock may be removed, and on every subsequent turn, one may remove either twice as many rocks or the same number of rocks as they have discarded on the previous turn. Determine the minimum number of turns required to remove exactly $2012$ rocks from the table.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Given the natural numbers $1,2,3,\ldots,10,11,12$, divide them into two groups such that the quotient of the product of all numbers in the first group by the product of all numbers in the second group is an integer and takes on the smallest possible value. What is this quotient?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{231}
Given the vertices of a regular 100-sided polygon \( A_{1}, A_{2}, A_{3}, \ldots, A_{100} \), in how many ways can three vertices be selected such that they form an obtuse triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["117600"]}
Eight coins are arranged in a circle heads up. A move consists of flipping over two adjacent coins. How many different sequences of six moves leave the coins alternating heads up and tails up?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7680}
Ava and Tiffany participate in a knockout tournament consisting of a total of $32$ players. In each of the $5$ rounds, the remaining players are paired uniformly at random. In each pair, both players are equally likely to win, and the loser is knocked out of the tournament. The probability that Ava and Tiffany play each other during the tournament is $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{116}
Find the maximum value of the function $f(x) = -4x^3 + 3x + 2$ over the interval $[0, 1]$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
For the odd function $f(x)$ defined on domain $\mathbb{R}$ that satisfies $f(4 - x) + f(x) = 0$, given that $f(x) = 2^x$ for $-2 < x < 0$, calculate $f(\log_2 20)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{4}{5}}
Let $S_i$ be the set of all integers $n$ such that $100i\leq n < 100(i + 1)$ . For example, $S_4$ is the set $400,401,402,\cdots,499$ . How many of the sets $S_0, S_1, S_2, \cdots, S_{999}$ do not contain a perfect square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{708}
An object is launched at 20 m/s from a height of 65 m. The equation for the height ($h$) in terms of time ($t$) is given by:

$$h(t) = -4.9t^2 + 20t + 65$$

What is the object's maximum height?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{85.4}
在锐角ΔABC中，$\angle A = 6 0 ^ { \circ } ,$ AB&gt;AC,O是外心，高BE，CF交于点H，M，N分别在线段BH，HF上，且满足BM=CN.求$\frac { M H + N H } { O H }$的值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\sqrt { 3 } $}
Exercise 5. A number has been written on the board. At each step, we add to it the largest of its digits (for example, if 142 is written, the next number will be 146). What is the maximum number of odd numbers that can be written consecutively by proceeding in this way?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
On the leg \(AC\) of a right triangle \(ABC\), a circle is constructed with \(AC\) as its diameter, intersecting the hypotenuse \(AB\) at point \(K\). Find the area of triangle \(CKB\), given that leg \(AC\) is equal to \(b\), and angle \(ABC\) is equal to \(\beta\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2} b^2 \cos^2 \beta \cot \beta}
Two circles touch each other externally at point \( A \). Through point \( B \) on their common tangent line \( AB \), two lines are drawn. One of these lines intersects the first circle at points \( M \) and \( N \), and the other intersects the second circle at points \( P \) and \( Q \). Given that \( AB = 6 \), \( BM = 9 \), and \( BP = 5 \), find the ratio of the areas of triangles \( MNO \) and \( PQO \), where point \( O \) is the intersection point of lines \( MP \) and \( NQ \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{625}{121}}
Given points $A$, $B$, and $C$ on circle $O$, if $\overrightarrow{AO} = \frac{1}{2}\left( \overrightarrow{AB} + \overrightarrow{AC} \right)$, find the angle between $\overrightarrow{AB}$ and $\overrightarrow{AC}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^{\circ}}
Let $Q$ be a $(2n+1) \times (2n+1)$ board. Some of its cells are colored black in such a way that every $2 \times 2$ board of $Q$ has at most $2$ black cells. Find the maximum amount of black cells that the board may have.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n^2+2n+1}
The numbers from 1 to 9 are placed in the cells of a \(3 \times 3\) table such that the sum of the numbers on one diagonal equals 7, and the sum on the other diagonal equals 21. What is the sum of the numbers in the five shaded cells?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Given that the perimeter of a sector is 10cm, and its area is 4cm<sup>2</sup>, find the radian measure of the central angle $\alpha$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{1}{2}"]}
Given the trinomial $16x^{2}+mxy+25y^{2}$, find the value of $m$ that makes it a perfect square trinomial. Express your answer as a numerical value, including the ± symbol if applicable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±40}
A sphere is inscribed in a cone. What maximum fraction of the cone's volume does the sphere occupy?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
设$A _ { 1 } ,$ $A _ { 2 } , \cdots ,$ $A _ { n }$都是9元集合{1,2,⋯,9}的子集，已知$\vert A _ { i } \vert$为奇数， $1\leq i\leq n,\vert A _ { i } \cap A _ { j } \vert$为偶数， $1\leq i\neq j\leq n$，则n的最大值为
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
How many ordered pairs of positive real numbers $(a,b)$ satisfy the equation:
\[(1+2a)(2+2b)(2a+b) = 32ab\]?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Cassandra sets her watch to the correct time at noon. At the actual time of 1:00 PM, she notices that her watch reads 12:57 and 36 seconds. Assuming that her watch loses time at a constant rate, calculate the actual time when her watch first reads 10:00 PM. Express your answer in the format HH:MM (12-hour clock), including AM/PM designation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10:25 PM}
Pasha wrote the numbers $4, 5, 6, \ldots, 14$ on the board and then erased one or more of them. It turned out that the remaining numbers on the board could not be divided into several groups such that the sums of the numbers in the groups were equal. What is the maximum possible value of the sum of the remaining numbers on the board?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{91}
Given the function $f(x) = x^2 - a \ln x$ with two roots, determine the range of values for $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2e, +\infty)}
In a certain World Cup, the Brazil team's probabilities of outcomes against each opponent are as follows:  
- Winning: $\frac{1}{2}$  
- Drawing: $\frac{1}{3}$  
- Losing: $\frac{1}{6}$  

Points awarded are:  
- Win: 3 points  
- Draw: 1 point  
- Loss: 0 points  

During the group stage, each team plays three matches, and a team needs to earn at least 4 points to advance to the knockout stage. In the knockout stage, if a match ends in a draw, it goes to a penalty shootout. The probability of Brazil winning a penalty shootout is $\frac{3}{5}$. What is the probability that Brazil wins the championship with exactly one of the four knockout stage matches decided by a penalty shootout?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{12}}
Given in $\bigtriangleup ABC$, $AB = 75$, and $AC = 120$. A circle with center $A$ and radius $AB$ intersects $\overline{BC}$ at points $B$ and $X$. Moreover, $\overline{BX}$ and $\overline{CX}$ have integer lengths. Find the length of $BC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{117}
Given a prime number \( p > 3 \), the sequence \( a_1, a_2, \cdots, a_{p-1} \) is a permutation of \( 1, 2, \cdots, p-1 \) such that
\[ p \mid (a_1 a_2 + a_2 a_3 + \cdots + a_{p-2} a_{p-1}). \]

Let the number of such permutations be \( K \). Find the remainder of \( K \) modulo \( p(p-1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p-1}
Compute the volume of the solid formed by rotating the astroid $x = a \cos^{3}(t), y = a \sin^{3}(t)$ around the line $y = -a$, where $a > 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{4} \pi^2 a^3}
For how many $x \in \{1,2,3,\dots, 2024\}$ is it possible that Bekhzod summed $2024$ non-negative consecutive integers, Ozod summed $2024+x$ non-negative consecutive integers and they got the same result?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1138}
Count the number of $x$ in $\{1, \ldots, n\}$ such that $x^{2} \equiv 1 \pmod{n}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the sum: $\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} \frac{m^{2} n}{3^{m}\left(n \cdot 3^{m}+m \cdot 3^{n}\right)}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9}{32}}
Given the function $f(x) = \lg \frac{\sum_{i=1}^{n-1} i^{x} + n^{x}a}{n}$, where $a \in \mathbb{R}$, for any positive integer $n \geqslant 3$, if the inequality $f(x) > (x-1)\lg n$ has a solution in the interval $[1, +\infty)$, then the range of values for the real number $a$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, +\infty)}
A right triangle \(ABC\) is divided by the altitude \(CD\), drawn to the hypotenuse, into two triangles \(BCD\) and \(ACD\). The radii of the circles inscribed in triangles \(BCD\) and \(ACD\) are 4 cm and 3 cm, respectively. Determine the distance between their centers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5\sqrt{2} \text{ cm}}
Find the greatest integer less than or equal to \(\sum_{k=1}^{2^{13a}} k^{1/1983 - 1}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1983}
In $\triangle ABC$, $AB=6$, $AC=8$, $BC=10$, and $D$ is the midpoint of $\overline{BC}$. What is the sum of the radii of the circles inscribed in $\triangle ADB$ and $\triangle ADC$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{17}{6}$}
Let $S = \{1,2,\cdots,2013\}$. Let $N$ denote the number of $9$-tuples of sets $(S_1, S_2, \dots, S_9)$ such that $S_{2n-1}, S_{2n+1} \subseteq S_{2n} \subseteq S$ for $n=1,2,3,4$. Find the remainder when $N$ is divided by $1000$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{369}
Let $P=\left\{x\mid x+2\geqslant x^{2}\right\}$, $Q=\left\{x\in \mathbb{N}\mid |x|\leqslant 3\right\}$. Calculate the intersection of sets P and Q, and express your answer as a set of numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left\{0, 1, 2\right\}}
Find all  $t\in \mathbb Z$  such that: exists a function  $f:\mathbb Z^+\to \mathbb Z$  such that: $f(1997)=1998$  $\forall x,y\in \mathbb Z^+ , \text{gcd}(x,y)=d : f(xy)=f(x)+f(y)+tf(d):P(x,y)$ 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ t = -1 }
In the plane rectangular coordinate system \(xOy\), \(F\) is the focus of the parabola \(\Gamma: y^2 = 2px (p > 0)\). Point \(B\) is on the \(x\)-axis, and is to the right of point \(F\). Point \(A\) is on \(\Gamma\), and \(|AF| = |BF|\). The second intersection points of the line \(AF\) and the line \(AB\) with \(\Gamma\) are \(M\) and \(N\) respectively. If \(\angle AMN = 90^\circ\), find the slope of the line \(AF\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
There are three piles of stones. Each time A moves one stone from one pile to another, A earns a reward from B. The reward is equal to the difference in the number of stones between the pile to which A moves the stone and the pile from which A takes the stone. If this difference is negative, A must pay B that amount instead (if A doesn’t have enough money, A can owe it temporarily). At a certain moment, all the stones are in the piles where they initially started. Determine the maximum possible amount of money A can have by that moment.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
The sum \( \sum_{n=0}^{2016\cdot2017^2}2018^n \) can be represented uniquely in the form \( \sum_{i=0}^{\infty}a_i\cdot2017^i \) for nonnegative integers \( a_i \) less than 2017. Compute \( a_0+a_1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the sum of the $x$-coordinates of the solutions to the system of equations $y=|x^2-8x+12|$ and $y=4-x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Let  $m,n > 2$  be even integers. Consider a board of size  $m \times n$  whose every cell is colored either black or white. The Guesser does not see the coloring of the board but may ask the Oracle some questions about it. In particular, she may inquire about two adjacent cells (sharing an edge) and the Oracle discloses whether the two adjacent cells have the same color or not. The Guesser eventually wants to find the parity of the number of adjacent cell-pairs whose colors are different. What is the minimum number of inquiries the Guesser needs to make so that she is guaranteed to find her answer?(Czech Republic)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m + n - 4}
$A$，$B$，$C$为$\Delta ABC$的三个内角，且$ctg\frac{A}{2}+ctg\frac{B}{2}+ctg\frac{C}{2}-2(ctgA+ctgB+ctgC)\ge T$.则${{T}_{\max }}=$_____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\sqrt{3}$.}
Find the value of $\displaystyle \int^{\pi}_{0}\cos^4(x+\sin 3x)dx$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{3\pi}{8}\)}
A wooden cube has edges of length 3 meters. Square holes, of side one meter, centered in each face are cut through to the opposite face. The edges of the holes are parallel to the edges of the cube. Calculate the entire surface area of the cube, including the inside surfaces exposed by the holes. Express your answer in square meters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
Given a hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$ with left and right foci $F\_1$, $F\_2$, and a focal distance of $2c (c > 0)$. If a parabola $y^{2} = 4cx$ intersects the hyperbola in the first quadrant at point $M$, and $|MF\_1| = 4c$, find the eccentricity of the hyperbola.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + \frac{\sqrt{6}}{3}}