inclusionAI/Ring-lite-rl-data huggingface dataset:
Let $d_1$ and $d_2$ be parallel lines in the plane. We are marking $11$ black points on $d_1$, and $16$ white points on $d_2$. We are drawing the segments connecting black points with white points. What is the maximum number of points of intersection of these segments that lies between the parallel lines (excluding the intersection points on the lines)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6600}
For which real numbers $a$, there exist pairwise different real numbers $x, y, z$ satisfying
$$\frac{x^3+a}{y+z}=\frac{y^3+a}{x+z}=\frac{z^3+a}{x+y}= -3.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ a \in (-2, 2) \setminus \{0\} }
Evaluate the limit: \[ \lim_{t\rightarrow 0}\left[-t^{-4} + t^{-5}\left(1+\frac{t^2}{3}\right)\tan^{-1}t\right] \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{45}}
Tickets for a basketball tournament were $6 for students and $9 for nonstudents. 250 more student tickets were sold than nonstudents, and 850 tickets of each type were sold. What was the total sales amount for the basketball tournament?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$14,250$}
Origami. Mitya plans to fold a square sheet of paper \(ABCD\). Mitya calls a fold beautiful if side \(AB\) intersects side \(CD\) and the four resulting right triangles are equal.

Before this, Vanya selects a random point \(F\) on the sheet. Find the probability that Mitya can make a beautiful fold that passes through point \(F\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
You roll a fair 12-sided die repeatedly. The probability that all the prime numbers show up at least once before seeing any of the other numbers can be expressed as a fraction \( \frac{p}{q} \) in lowest terms. What is \( p+q \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["793"]}
Given a lawn with a rectangular shape of $120$ feet by $200$ feet, a mower with a $32$-inch swath width, and a $6$-inch overlap between each cut, and a walking speed of $4000$ feet per hour, calculate the time it will take John to mow the entire lawn.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.8}
Four disks with disjoint interiors are mutually tangent. Three of them are equal in size and the fourth one is smaller. Find the ratio of the radius of the smaller disk to one of the larger disks.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{2\\sqrt{3} - 3}{3}"]}
Eight unmarked envelopes contain the numbers: 1, 2, 4, 8, 16, 32, 64, 128. Eve chooses a few envelopes randomly. Alie takes the rest. Both sum up their numbers. Eve's sum is 31 more than Alie's. Find the number of envelopes that Eve took. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
10. A sphere touches all the edges of the pyramid $S A B C$, and the lateral edges $S A, S B$, and $S C$ at points $A^{\prime}, B^{\prime}$, and $C^{\prime}$. Find the volume of the pyramid $S A^{\prime} B^{\prime} C^{\prime}$, if $A B=B C=S B=5$ and $A C=4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2\sqrt{59}}{15}}
James decides to cut down some trees. In the first 2 days, he cuts down 20 trees each day. For the next 3 days, his 2 brothers help him cut down trees. They cut 20% fewer trees per day than James. Then, their cousin joins them for the following 4 days; the cousin manages to cut down 15% more trees per day than James. Finally, they hire a professional woodcutter who has a daily productivity of 30 trees. They all work together for the next 5 days. How many total trees were cut down during this entire period?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1021}
Calculate the number of distinct (unordered) hands that can be dealt to a player from a deck of 52 cards when choosing 10 cards. Express your answer as a single digit, which represents the digit A in the number 158A00A4AA0.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
设 S=\int_{-\infty}^{+\infty}\df{\dd x}{1+x^2+x^4}，求 [100S^2].
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{328}
A triangle is divided into 1000 triangles. What is the maximum number of distinct points that can be vertices of these triangles?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1002}
The height of a rhombus, drawn from the vertex of its obtuse angle, divides the side of the rhombus in the ratio $1:3$ as measured from the vertex of its acute angle. What fraction of the rhombus's area is the area of the circle inscribed in it?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi \sqrt{15}}{16}}
If the price of a book is first decreased by 25%, then increased by 20%, after which it is decreased by 15% and finally increased by 30%, then the net change in the price will be:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.55\%}
A train of a certain length is running with a speed of 55 km/hr. In 10.45077684107852 seconds, it will pass a man who is running at 7 km/hr in the direction opposite to that in which the train is going. What is the length of the train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{180}
Determine the number of subgroups of order $q$ in a finite abelian $q$-group, where $q$ is a prime number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{q^n-1}{q-1}\)}
Find the number of ordered integer pairs \((a, b)\) such that the equation \(x^{2} + a x + b = 167y\) has integer solutions \((x, y)\), where \(1 \leq a, b \leq 2004\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2020032}
Given the function $f(x) = x^{n-1}\ln(x)$, determine the expression for its $n$-th derivative $f^{(n)}(x)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{(n-1)!}{x}}
In $\triangle ABC$, the angles opposite to sides $a$, $b$, $c$ are $A$, $B$, $C$, respectively. If $\sin^{2}A+\sin^{2}B-\sin^{2}C=\sin A\sin B$, find the measure of angle $C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac {\pi}{3}}
If  $x = cos 1^o cos 2^o cos 3^o...cos 89^o$  and  $y = cos 2^o cos 6^o cos 10^o...cos 86^o$ , then what is the integer nearest to  $\frac27 \log_2 \frac{y}{x}$  ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Consider the equation: 250,957 + x^3 = 18,432,100. What is the value of x?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{263}
Among the positive integers less than $10^{4}$, how many positive integers $n$ are there such that $2^{n} - n^{2}$ is divisible by 7?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["2857"]}
In a square $A B C D$ with side length 2, $E$ is the midpoint of $A B$. The square is folded along segments $E C$ and $E D$ such that segments $E A$ and $E B$ coincide (point $B$ coincides with point $A$), forming the tetrahedron $C D E A$. Find the volume of this tetrahedron.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{3}}
Ms. Math's kindergarten class has $16$ registered students. The classroom has a very large number, $N$ , of play blocks which satisfies the conditions: 
$(a)$ If $16$ , $15$ , or $14$ students are present in the class, then in each case all the blocks can be distributed in equal numbers to each student, and $(b)$ There are three integers $0 < x < y < z < 14$ such that when $x$ , $y$ , or $z$ students are present and the blocks are distributed in equal numbers to each student, there are exactly three blocks left over. 
Find the sum of the distinct prime divisors of the least possible value of $N$ satisfying the above conditions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{148}
In a rhombus $P Q R S$ with $P Q=Q R=R S=S P=S Q=6$ and $P T=R T=14$, what is the length of $S T$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Find the number of permutations of $\{1,2,...,n\}$ like $\{a_1,...,a_n\}$ st for each $1 \leq i \leq n$:
$$a_i | 2i$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ 2^{\lfloor n/2 \rfloor} }
Determine the values of \( \alpha \) for which the integral \( \int_{1}^{\infty}\left(\sum_{j = 1}^{10}x^{j}\right)^{-(n +1)/10}\,\frac{-n + 1}{10}\,x^{\alpha}\,\mathrm{d}x \) is convergent, given that \( n > 0 \) and \( \alpha \) is a real number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha < n}
Let $S$ be the set $\{1,2,3,...,19\}$. For $a,b \in S$, define $a \succ b$ to mean that either $0 < a - b \le 9$ or $b - a > 9$. How many ordered triples $(x,y,z)$ of elements of $S$ have the property that $x \succ y$, $y \succ z$, and $z \succ x$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$855$}
Points \( C \) and \( D \) have the same \( y \)-coordinate of 25, but different \( x \)-coordinates. What is the difference between the slope and the \( y \)-intercept of the line containing both points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-25}
Three players are playing table tennis, where the player who loses a game gives up their place to a player who was not participating. It turns out that the first player played 10 games and the second player played 21 games. How many games did the third player play?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
A circle can be inscribed in the quadrilateral \(ABCD\). Let \(K\) be the intersection point of its diagonals. It is known that \(AB > BC > KC\), \(BK = 4 + \sqrt{2}\), and the perimeter and area of triangle \(BKC\) are 14 and 7, respectively.

Find \(DC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Consider a circle $O$ with an inscribed triangle $\triangle ABC$. Let the midpoints of arcs $\overparen{AB}$, $\overparen{BC}$, and $\overparen{CA}$ be $C_1$, $A_1$, and $B_1$ respectively, forming an inscribed triangle $\triangle A_1B_1C_1$. Then, take the midpoints of arcs $\overparen{A_1B_1}$, $\overparen{B_1C_1}$, and $\overparen{C_1A_1}$ as $C_2$, $A_2$, and $B_2$ respectively, forming another inscribed triangle $\triangle A_2B_2C_2$. Determine the behavior of the angles of $\triangle A_nB_nC_n$ as $n$ approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
Evaluate $$\lim_{n \to \infty} \sum_{k=1}^n \left[\left(\cos\left(\frac{2\pi(k+1)}n \right) +\cos\left(\frac{2\pi k}n\right) \right) \left(\sin\left(\frac{2\pi(k+1)}n \right) -\sin\left(\frac{2\pi k}n\right) \right) \right]$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2π}
Evaluate the integral $$\iint_R\sqrt \frac{x+y}{x-2y}dA$$ where $R$ is the region bounded by $ y - \frac{x}{2} = 0, y = 0,x+y = 1.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/3}
A circle passing through the vertices $A$, $B$, and $C$ of the parallelogram $ABCD$ is tangent to line $AD$ and intersects line $CD$ at points $C$ and $M$. Find the length of segment $AD$, if $BM = 9$ and $DM = 8$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6\sqrt{2}}
Find the smallest number in which all digits are different and the sum of all digits equals 32.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26789}
Given sets $P = \{1, 5, 10\}$ and $S = \{1, 3, a^2+1\}$, if the union of $S$ and $P$ is $S \cup P = \{1, 3, 5, 10\}$, find the value of the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a = \pm 3}
To investigate a non-luminous black planet in distant space, Xiao Feitian drives a high-speed spaceship equipped with a powerful light, traveling straight towards the black planet at a speed of 100,000 km/s. When Xiao Feitian had just been traveling for 100 seconds, the spaceship instruments received light reflected back from the black planet. If the speed of light is 300,000 km/s, what is the distance from Xiao Feitian's starting point to the black planet in 10,000 kilometers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2000}
Determine the initial value to be used when applying the method of characteristics to solve the partial differential equation: \[ u_t + (1-x)u_x = 0, \quad x \in [0,1], \ t>0 \] with the initial condition \( u(x,0) = g(x), \ x \in [0,1] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{g\left(1 + (x - 1)e^{t}\right)}
Let $n \geq 3$ be an integer. Rowan and Colin play a game on an $n \times n$ grid of squares, where each square is colored either red or blue. Rowan is allowed to permute the rows of the grid and Colin is allowed to permute the columns. A grid coloring is [i]orderly[/i] if: [list] [*]no matter how Rowan permutes the rows of the coloring, Colin can then permute the columns to restore the original grid coloring; and [*]no matter how Colin permutes the columns of the coloring, Rowan can then permute the rows to restore the original grid coloring. [/list] In terms of $n$, how many orderly colorings are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2(n! + 1)}
7

An engineer arrives at the train station at 8 o'clock in the morning every day. At exactly 8 o'clock, a car arrives at the station and takes the engineer to the factory. One day, the engineer arrived at the station at 7 o'clock and started walking towards the car. Meeting the car, he got in and arrived at the factory 20 minutes earlier than usual. How long did the engineer walk? The speeds of the car and the engineer are constant.

#
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
$$
\begin{array}{l}
\text { Find }  \lim_{n \to \infty}  a_{n} \text { where }\left(a_{n}\right)_{n \geq 1} \text { is defined by }\\
a_{n}=\frac{1}{\sqrt{n^{2}+8 n-1}}+\frac{1}{\sqrt{n^{2}+16 n-1}}+\frac{1}{\sqrt{n^{2}+24 n-1}}+\ldots+\frac{1}{\sqrt{9 n^{2}-1}}
\end{array}
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\[ \frac{1}{2} \]}
Determine the number of group structures that make the unit circle $S^1$ a topological group.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the smallest integer $n$ such that the Simpson's Rule approximation for the integral \( \int_0^1 9e^{x^2} \, dx \) is accurate to within 0.0001.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Pat Peano has plenty of 0's, 1's, 3's, 4's, 5's, 6's, 7's, 8's and 9's, but he has only twenty-two 2's.  How far can he number the pages of his scrapbook with these digits?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{119}
Determine the number of terms, $n$, required in the series expansion of $\pi = 4\arctan(1/2) + 4\arctan(1/3)$ such that the error is less than $10^{-10}$. Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
An $10 \times 25$ rectangle is divided into two congruent polygons, and these polygons are rearranged to form a rectangle again. Determine the length of the smaller side of the resulting rectangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
On the extensions of the sides of triangle $ABC$, points $A_{1}$, $B_{1}$, and $C_{1}$ are taken such that $\overrightarrow{A B_{1}} = 2 \overrightarrow{A B}$, $\overrightarrow{B C_{1}} = 2 \overrightarrow{B C}$, and $\overrightarrow{C A_{1}} = 2 \overrightarrow{A C}$. If the area of triangle $ABC$ is $S$, find the area of triangle $A_{1} B_{1} C_{1}$. Express your answer in terms of $S$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5S}
Max repeatedly throws a fair coin in a hurricane. For each throw, there is a $4 \%$ chance that the coin gets blown away. He records the number of heads $H$ and the number of tails $T$ before the coin is lost. (If the coin is blown away on a toss, no result is recorded for that toss.) What is the expected value of $|H-T|$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{24}{7}}
Given the equation $(t+1)\cos x-t\sin x=t+2$ has a real root in $(0,\pi)$, the maximum value of the real number $t$ is \_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
A sports team in a certain class consists of 18 soccer players, 12 basketball players, and 6 table tennis players (each person participates in only one sport). Now, a sample of size $n$ is drawn from these athletes. If both systematic sampling and stratified sampling methods are used, no individuals need to be excluded; when the sample size is $n+1$, if systematic sampling is used, 1 individual needs to be excluded. Then, the sample size $n$ is \_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
An obtuse triangle with positive area has side lengths 10, 17, and \(x\). If \(x\) is an integer, what is the sum of all possible values of \(x\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{224}
A spherical soap bubble lands on a horizontal wet surface and forms a hemisphere. The volume of the hemisphere is known to be $36\pi$ cm³. Find the radius of the original bubble.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 \text{ cm}}
Let $D$ be a point on side $[AB]$ of triangle $ABC$ with $|AB| = |AC|$, such that $[CD]$ is an angle bisector and $m(\widehat{ABC}) = 40^\circ$. Let $F$ be a point on the extension of $[AB]$ beyond $B$ such that $|BC| = |AF|$. Let $E$ be the midpoint of $[CF]$. If $G$ is the intersection of lines $ED$ and $AC$, what is the value of $x$ if $m(\widehat{FBG}) = x^\circ$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}
A unit cube has vertices \(P_1, P_2, P_3, P_4, P_1', P_2', P_3'\), and \(P_4'\). Vertices \(P_2, P_3\), and \(P_4\) are adjacent to \(P_1\), and for \(1\leq i\leq 4\), vertices \(P_i\) and \(P_i'\) are opposite to each other. A regular octahedron has one vertex in each of the segments \(P_1P_2, P_1P_3, P_1P_4, P_1'P_2', P_1'P_3'\), and \(P_1'P_4'\). Find the octahedron's side length. The original answer is in the form \(\frac{a\sqrt{b}}{c}\), please give the value of \(a + b + c\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
3. Find the minimum value of the expression

$$
A=\left(5\left(\cos x_{1}+\ldots+\cos x_{n}\right)+\sin x_{1}+\ldots+\sin x_{n}\right) \cdot\left(\cos x_{1}+\ldots+\cos x_{n}-5\left(\sin x_{1}+\ldots+\sin x_{n}\right)\right)
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-13n^{2}}
Miki has a dozen oranges of the same size and a dozen pears of the same size. Miki uses her juicer to extract 8 ounces of pear juice from 3 pears and 8 ounces of orange juice from 2 oranges. She makes a pear-orange juice blend from an equal number of pears and oranges. Calculate the percentage of the blend that is pear juice. Express your answer as a percentage (%).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40%}
Given $m\in R$, the distance from point $P(x,y)$ where the line $x+my=0$ intersects with the line $mx-y-2m+4=0$ to the line $l:(x-1)\cos θ+(y-2)\sin θ=3$ is maximized at $\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3+\sqrt{5}}
A buyer took goods worth 10 rubles from a seller and gave 25 rubles. The seller did not have change, so he exchanged money with a neighbor. After they finished the transaction and the buyer left, the neighbor discovered that the 25 rubles were counterfeit. The seller returned 25 rubles to the neighbor and started thinking. What loss did the seller incur?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25 \, \text{rubles}}
Sequence \(a_{1}, a_{2}, \cdots\) is defined as follows: \(a_{n}=2^{n}+3^{n}+6^{n}-1\), for \(n=1,2,3, \cdots\). Find all positive integers that are relatively prime to every term of this sequence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In triangle $ABC,$ $AB = 20$ and $BC = 15.$  Find the largest possible value of $\tan A.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3 \sqrt{7}}{7}}
The integers from 1 to $2008^2$ are written on each square of a $2008 \times 2008$ board. For every row and column, the difference between the maximum and minimum numbers is computed. Let $S$ be the sum of these 4016 numbers. Find the greatest possible value of $S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16184704896}
How many distinct sets of $5$ distinct positive integers $A$ satisfy the property that for any positive integer $x \leq 29$, a subset of $A$ sums to $x$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Alyssa spent half of her allowance at the movies and 20% on snacks. She saved 1/4 of her allowance for a future purchase. She washed the family car and earned 8 dollars and also mowed the lawn for 5 dollars. If she ended up with 12 dollars after spending on movies and snacks, and saving for the future purchase, what is her weekly allowance?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{240}
Determine the radius of convergence for the series: $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \left(\left|\frac{x-1}{x+1}\right|-1\right)^n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
It is known that in triangle $ABC$, $AB = a$, and $\angle C = \alpha$. Find the radius of the circle passing through vertices $A$, $B$, and the center of the circle inscribed in triangle $ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a}{2 \cos \frac{\alpha}{2}}}
11. From the three-digit numbers $100, 101, 102, \ldots, 699, 700$, if $n$ different numbers are taken, such that there are always three numbers among them with the same sum of digits. Then the minimum value of $n$ is $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{47}
在RtΔABC中，$\angle C = 9 0 ^ { \circ } , A C = 1 1$，点D在边AC上且AD=10,∠BDC=3∠BAC..若将 ΔABC的周长写成$\frac { 1 } { 2 } ( m + \sqrt { n } )$ 的形式（m，n为整数），则m+n= ·
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{638}
Given that $x$ and $y$ are independent standard Gaussian random variables, calculate the conditional probability $P(x > 0 \mid x > y)$. Provide your answer as a numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{4}}
Find $B^2$, where $B$ is the sum of the absolute values of all roots of the equation:
\[x = \sqrt{26} + \frac{119}{{\sqrt{26}+\frac{119}{{\sqrt{26}+\frac{119}{{\sqrt{26}+\frac{119}{{\sqrt{26}+\frac{119}{x}}}}}}}}}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{502}