inclusionAI/Ring-lite-rl-data huggingface dataset:
A school provides apples, bananas, and pears as snacks for students. Each student must choose at least one fruit, but can select multiple. The statistics show that 70% of the students choose apples, 40% choose bananas, and 30% choose pears. What is the maximum percentage of students who chose all three fruits?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20\%}
A quadrilateral is divided into 1000 triangles. What is the maximum number of distinct points that can be the vertices of these triangles?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1002}
The sum of one hundred numbers is 1000. The largest of them was doubled, and another number was decreased by 10. It turned out that the sum did not change. Find the smallest of the original numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Evaluate the line integral \( \int_C F \cdot dr \) where \( F = (5xy^3)i + (3x^2y^2)j \) along the curve \( C \) consisting of the x-axis from \( x=0 \) to \( x=4 \), the parabola \( y=16-x^2 \) up to the y-axis, and the y-axis down to the origin.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24576}
One angle of an isosceles triangle is 30 degrees. Find the possible difference between the other two angles. Express your answer in degrees as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
Find the flux of the vector field \( \mathbf{a} = \mathbf{i} - \mathbf{j} + xyz \mathbf{k} \) through the circle \( S \), obtained by the intersection of the sphere \( x^2 + y^2 + z^2 \leq R^2 \) with the plane \( y = x \). Take the side of the circle facing the positive part of the \( O_x \)-axis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi R^{2}}
The random variable $X$ follows a normal distribution $N(1, 4)$, where the mean $\mu = 1$ and the variance $\sigma^2 = 4$. Given that $P(X \geq 2) = 0.2$, calculate the probability $P(0 \leq X \leq 1)$. Express your answer as a decimal value rounded to 1 decimal place.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.3}
When studying the operation of a new type of cyclic thermal engine, it was found that during part of the period it receives heat, and the absolute power of heat supply is expressed by the law: 
\[ P_{1}(t)=P_{0} \frac{\sin (\omega t)}{100+\sin (t^{2})}, \quad 0<t<\frac{\pi}{\omega}. \] 
The gas performs work, developing mechanical power 
\[ P_{2}(t)=3 P_{0} \frac{\sin (2 \omega t)}{100+\sin (2 t)^{2}}, \quad 0<t<\frac{\pi}{2 \omega}. \] 
The work on the gas performed by external bodies is \( \frac{2}{3} \) of the work performed by the gas. Determine the efficiency of the engine.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
. Let $n$ be an integer with $n \geq 3$. Consider all dissections of a convex $n$-gon into triangles by $n-3$ non-intersecting diagonals, and all colourings of the triangles with black and white so that triangles with a common side are always of a different colour. Find the least possible number of black triangles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\lfloor\frac{n-1}{3}\rfloor}
Calculate the Laplacian of $\frac{z}{s}$, where $s = \sqrt{x^2 + y^2 + z^2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\,\frac{2z}{s^{3}}}
Let $q(z)=z^5+z(z-2)^2+1$. Find the number of zeros of $q(z)$ in the disk $|z-2|<1$, considering multiplicities.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Using systematic sampling, extract a sample of size 12 from a population of 123 individuals. The sampling interval is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
The Absent-Minded Scientist commutes to work on weekdays using the Ring Line of the Moscow Metro from "Taganskaya" station to "Kievskaya" station, and back in the evening. When he enters the station, he boards the first arriving train. It is known that in both directions, trains run at approximately equal intervals. The train traveling via the northern route (through "Belorusskaya") takes 17 minutes to go from "Kievskaya" to "Taganskaya" or vice versa, while the train traveling via the southern route (through "Paveletskaya") takes 11 minutes.

According to his long-term observations, the train traveling counterclockwise arrives at "Kievskaya" on average 1 minute and 15 seconds after the clockwise train arrives. The same is true for "Taganskaya." Moreover, the scientist notes that his average travel time from home to work is 1 minute less than his average travel time from work to home. Find the expected interval between trains traveling in one direction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 \text{ minutes}}
7. Problem: Let $a, b, c, d$ be integers such that $a>b>c>d \geq-2021$ and
$$
\frac{a+b}{b+c}=\frac{c+d}{d+a}
$$
(and $b+c \neq 0 \neq d+a)$. What is the maximum possible value of $a c$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{510050}
What is the  $33$ rd number after the decimal point of $(\sqrt{10} + 3)^{2001}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
设 f(x) 是定义域为 \bR、最小正周期为 5 的函数. 若函数 g(x)=f(2^x) 在区间 [0,5) 上的零点个数为 25，则 g(x) 在区间 [1,4) 上的零点个数为__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Gunner has a box of mittens and a box of plugs. The box of mittens has 150 pairs of mittens, and he decides to put 30 more pairs of plugs into the box of plugs. Now, there are 400 plugs. How many fewer pairs of mittens are there compared to the original number of pairs of plugs?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Determine the number of ring isomorphisms from $\mathbb{Z}^n$ to $\mathbb{Z}^n$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n!}
For which values of the parameter \( a \), the equation

\[ 
\log _{2}\left(2 x^{2}-x-2 a-4 a^{2}\right)+3 \log _{1 / 8}\left(x^{2}-a x-2 a^{2}\right)=0 
\]

has two distinct roots, the sum of whose squares belongs to the interval \( (4, 8) \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{( \frac{3}{5}, 1 )}
Evaluate the improper integral \( \int_1^\infty \frac{1}{y^2}\left(\left\lfloor \beta y\right\rfloor-\beta\left\lfloor y\right\rfloor\right) \mathrm{d}y \) for \( \beta \in (0, 1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\beta \ln \beta}
The work team was working at a rate fast enough to process $1250$ items in ten hours. But after working for six hours, the team was given an additional $165$ items to process. By what percent does the team need to increase its rate so that it can still complete its work within the ten hours?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33\%}
A sequence is recursively defined as: \( t_{1} = 1 \). For \( n > 1 \), if \( n \) is even, \( t_{n} = 1 + t_{\frac{n}{2}} \); if \( n \) is odd, \( t_{n} = \frac{1}{t_{n-1}} \). Given that \( t_{n} = \frac{19}{87} \), find the sum of the digits of \( n \). Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
An infinite sequence of integers, $a_0, a_1, a_2, \dots$, with $a_0 > 0$, has the property that for $n \ge 0$, $a_{n+1} = a_n - b_n$, where $b_n$ is the number having the same sign as $a_n$, but having the digits written in reverse order. For example, if $a_0 = 1210$, $a_1 = 1089$, and $a_2 = -8712$, etc. Find the smallest value of $a_0$ such that $a_n \neq 0$ for all $n \ge 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1012}
In a game involving the polynomial equation $x^3 + Ax^2 + Bx + C = 0$, two players, Boy and Girl, take turns as follows: Boy chooses a real number, and Girl assigns it to one of the coefficients $A$, $B$, or $C$. This process is repeated until all coefficients are determined. If the resulting polynomial has three distinct integer roots, Boy wins; otherwise, Girl wins. Determine if either player has a winning strategy, and if so, identify the player.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Girl}
Let $T_k$ be the transformation of the coordinate plane that first rotates the plane $k$ degrees counterclockwise around the origin and then reflects the plane across the $y$-axis. What is the least positive integer $n$ such that performing the sequence of transformations $T_1, T_2, T_3, \cdots, T_n$ returns the point $(1,0)$ back to itself?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$359$}
An urn contains $w$ white and $b$ black balls. $n$ extractions without replacement are made. What is the conditional probability of drawing black on the $3$rd draw and white on the $6$th draw given that a black ball was drawn on the $2$nd draw and a white ball on the $4$th draw?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{(b-1)(w-1)}{(w+b-2)(w+b-3)}}
Let \( a, b, c \in \mathbb{R}^{+} \) and \( a b c + a + c = b \). Determine the maximum value of \( P = \frac{2}{a^{2} + 1} - \frac{2}{b^{2} + 1} + \frac{3}{c^{2} + 1} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{10}{3}}
Example 10 Let real numbers $x, y, z, w$ satisfy
$$
\left\{\begin{array}{l}
\frac{x^{2}}{2^{2}-1^{2}}+\frac{y^{2}}{2^{2}-3^{2}}+\frac{z^{2}}{2^{2}-5^{2}}+\frac{w^{2}}{2^{2}-7^{2}}=1, \\
\frac{x^{2}}{4^{2}-1^{2}}+\frac{y^{2}}{4^{2}-3^{2}}+\frac{z^{2}}{4^{2}-5^{2}}+\frac{w^{2}}{4^{2}-7^{2}}=1, \\
\frac{x^{2}}{6^{2}-1^{2}}+\frac{y^{2}}{6^{2}-3^{2}}+\frac{z^{2}}{6^{2}-5^{2}}+\frac{w^{2}}{6^{2}-7^{2}}=1, \\
\frac{x^{2}}{8^{2}-1^{2}}+\frac{y^{2}}{8^{2}-3^{2}}+\frac{z^{2}}{8^{2}-5^{2}}+\frac{w^{2}}{8^{2}-7^{2}}=1 .
\end{array}\right.
$$

Try to find the value of $x^{2}+y^{2}+z^{2}+w^{2}$.
(From the 2nd American Invitational Mathematics Examination)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Initially the equation
$$\star \frac{1}{x-1} \star \frac{1}{x-2} \star \frac{1}{x-4} ... \star \frac{1}{x-2^{2023}}=0$$
is written on the board. In each turn Aslı and Zehra deletes one of the stars in the equation and writes $+$ or $-$ instead. The first move is performed by Aslı and continues in order. What is the maximum number of real solutions Aslı can guarantee after all the stars have been replaced by signs?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
1011
}
Square  $CASH$  and regular pentagon  $MONEY$  are both inscribed in a circle. Given that they do not share a vertex, how many intersections do these two polygons have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Triangle $ABC$ with $\angle A = 90^\circ$ has incenter $I$. A circle passing through $A$ with center $I$ is drawn, intersecting $\overline{BC}$ at $E$ and $F$ such that $BE < BF$. If $\frac{BE}{EF} = \frac{2}{3}$, then $\frac{CF}{FE} = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Given a six-digit decimal number formed by six positive integers, where the unit digit is a multiple of 4, the tens and hundreds digits are multiples of 3, and the sum of the digits of the six-digit number is 21. How many six-digit numbers satisfy the above conditions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{126}
We draw a circle with radius $5$ on a gridded paper where the grid consists of squares with sides of length $1$. The center of the circle is placed in the middle of one of the squares. Find the number of squares through which the circle passes. Please provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Let $p=2^{16}+1$ be a prime. A sequence of $2^{16}$ positive integers $\{a_n\}$ is monotonically bounded if $1\leq a_i\leq i$ for all $1\leq i\leq 2^{16}$. We say that a term $a_k$ in the sequence with $2\leq k\leq 2^{16}-1$ is a mountain if $a_k$ is greater than both $a_{k-1}$ and $a_{k+1}$. Evan writes out all possible monotonically bounded sequences. Let $N$ be the total number of mountain terms over all such sequences he writes. Find the remainder when $N$ is divided by $p$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$49153$
}
In a kindergarten, each child was given three cards, each of which has either "MA" or "NY" written on it. It turned out that 20 children can arrange their cards to spell the word "MAMA", 30 children can arrange their cards to spell the word "NYANYA", and 40 children can arrange their cards to spell the word "MANYA". How many children have all three cards that are the same?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Suppose that $x$ is a real number such that \( \frac{27 \times 9^x}{4^x} = \frac{3^x}{8^x} \). Find the value of \( 2^{-(1+\log_2 3)x} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
There are 9 representatives from different countries, with 3 people from each country. They sit randomly around a round table with 9 chairs. What is the probability that each representative has at least one representative from another country sitting next to them?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{41}{56}}
If $a=( \frac {1}{2})^{ \frac {2}{3}}$, $b=( \frac {1}{5})^{ \frac {2}{3}}$, $c=( \frac {1}{2})^{ \frac {1}{3}}$, determine the relationship in size between $a$, $b$, and $c$ (connect the variables using the "<" symbol).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{b < a < c}
How many sets containing at least two elements of consecutive positive integers exist, where the sum of the elements is 100?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given a sequence ${a_n}$ where all terms are positive numbers, $a_1=2$, and $a_{n+1} - a_n = \frac{4}{a_{n+1}+a_n}$, if the sum of the first $n$ terms of the sequence ${\frac{1}{a_{n+1} + a_n}}$ is 5, calculate the value of $n$. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
A dog is being treated for fleas. The effectiveness of each flea treatment varies as follows: the first treatment eliminates 60% of fleas, the second treatment eliminates 45% of the remaining fleas, the third treatment eliminates 30% of the remaining fleas, the fourth treatment eliminates 20% of the remaining fleas, and the fifth treatment eliminates 10% of the remaining fleas. Additionally, the dog spends time with three other dogs, each of which has a different amount of fleas which could transfer to the first dog. The dog also spends time in an environment that could cause some fleas removed by treatment to return. After all five treatments, the dog has 25 fleas left. Taking into account these factors, how many more fleas did the dog have before the flea treatments than after?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{201}
Evaluate the integral \( \oint_{|w|=2} \frac{1}{w(5w^8-10w^4+4w^2+2iw+10)} \, dw \) given that all the zeros of the denominator lie within the circle \(|w|<1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
if 4 consecutive letters are selected at random from the english alphabet , then the probability that one of the letters is a vowel is ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18/23}
In triangle $ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. It is known that $\sin ^{2}B+\cos ^{2}A-\cos ^{2}C= \sqrt {3}\sin B\sin C$, and the area of the circumcircle of triangle $ABC$ is $4\pi$. Find the value of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
The function $y=\log_{\frac{1}{2}}(3x^2-ax+5)$ is decreasing on the interval $[-1,+\infty)$. The range of the real number $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-8, -6]}
How many 4x3 matrices of rank 3 can be formed where the entries are from a field with 3 elements?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{449280}
Ayu is about to open the suitcase but she forgets the key. The suitcase code consists of nine digits, namely four $0$s (zero) and five $1$s. Ayu remembers that no four consecutive numbers are the same. How many codes might have to try to make sure the suitcase is open?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{99}
Every pair of communities in a county are linked directly by one mode of transportation; bus, train, or airplane. All three methods of transportation are used in the county with no community being serviced by all three modes and no three communities being linked pairwise by the same mode. Determine the largest number of communities in this county.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let \(a, b, c, d\) be integers such that \(\frac{a}{c} \in \mathbb{Q}^+ \cap \mathbb{Z}^+\) and \(\frac{b}{d} \in \mathbb{Q}^- \cap \mathbb{Z}^-\). Determine the number of solutions to the equation \(|ad - bc| = 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
What is the name of the algebraic structure where the additive group of a ring is non-abelian, and only right distributivity is required?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{near-ring}
Six chairs sit in a row.  Six people randomly seat themselves in the chairs. Each person randomly chooses either to set their feet on the floor, to cross their legs to the right, or to cross their legs to the left.  There is only a problem if two people sitting next to each other have the person on the right crossing their legs to the left and the person on the left crossing their legs to the right.  The probability that this will **not** happen is given by  $\frac{m}{n}$  where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1106}
Find the smallest positive integer \( n \) such that in any 2-coloring of \( K_n \), there exist two monochromatic triangles of the same color that do not share any edge.
(1991 China National Training Team Problem)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
The side \( AD \) of the rectangle \( ABCD \) is equal to 2. On the extension of the side \( AD \) beyond point \( A \), a point \( E \) is taken such that \( EA = 1 \), and \( \angle BEC = 30^\circ \). Find \( BE \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given a regular pentagon \( A B C D E \). On side \( A E \), point \( K \) is marked and on side \( C D \), point \( L \) is marked. It is known that \( \angle L A E + \angle K C D = 108^\circ \) and \( \frac{A K}{K E} = \frac{3}{7} \). Find the ratio \( \frac{C L}{A B} \).

Note: A regular pentagon is a pentagon where all sides are equal and all angles are equal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.7}
In a pentagon \(ABCDE\) inscribed in a circle with radius \(R\), it is given that \(\angle B = 110^\circ\) and \(\angle E = 100^\circ\). Determine the length of side \(CD\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{R}
Consider the numbers 1 through 1000 arranged in a line. Each number, except the rightmost, must differ by 1 from at least one number to its right. Determine the total number of such possible arrangements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{999}}
You are given  $n$  not necessarily distinct real numbers  $a_1, a_2, \ldots, a_n$ . Let's consider all  $2^n-1$  ways to select some nonempty subset of these numbers, and for each such subset calculate the sum of the selected numbers. What largest possible number of them could have been equal to  $1$ ?

For example, if  $a = [-1, 2, 2]$ , then we got  $3$  once,  $4$  once,  $2$  twice,  $-1$  once,  $1$  twice, so the total number of ones here is  $2$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n-1}}
Evaluate the expression:
\[2 \times (2 \times (2 \times (2 \times (2 \times (2 \times 2 - 2) - 2) - 2) - 2) - 2) - 2.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the sum of the non-repeated roots of the polynomial $P(x) = x^6 - 5x^5 - 4x^4 - 5x^3 + 8x^2 + 7x + 7$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Determine the value of $P$ at which the function \( \frac{dP}{dt}=c P\left(\frac{\ln(k)}{P}\right) \) grows fastest, where \( c \) is a constant and \( k \) is a carrying capacity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{k}{e}}
Find the smallest positive integer \( k \) such that \( (k-10)^{5026} \geq 2013^{2013} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55}
In one year, what is the maximum number of months that can have five Sundays? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
The smallest number which, when diminished by 6, is divisible by some numbers including 12, 18, 21, and 28 is 1014. What is the second number that the diminished value is divisible by?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
4. All possible 6-digit numbers, in each of which the digits occur in non-increasing order (from left to right, e.g., 877550) are written as a sequence in increasing order. Find the 2005 -th number in this sequence.


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{864110}
Determine the lower bound of the radius of convergence for the power series \( \sum_{n \ge 1} a_n z^n \), where \( a_n \) is the number of divisors of \( n^{50} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let the circles $S_1$ and $S_2$ meet at the points $A$ and $B$. A line through $B$ meets $S_1$ at a point $D$ other than $B$ and meets $S_2$ at a point $C$ other than $B$. The tangent to $S_1$ through $D$ and the tangent to $S_2$ through $C$ meet at $E$. If $|AD|=15$, $|AC|=16$, $|AB|=10$, what is $|AE|$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Calculate the limit of the function:

\[
\lim_{{x \to 2}} \left(2 e^{x-2} - 1\right)^{\frac{3x+2}{x-2}}
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{16}}
Let \( OP \) be the diameter of the circle \( \Omega \), and \( \omega \) be a circle centered at point \( P \) with a radius smaller than that of \( \Omega \). The circles \( \Omega \) and \( \omega \) intersect at points \( C \) and \( D \). The chord \( OB \) of the circle \( \Omega \) intersects the second circle at point \( A \). Find the length of segment \( AB \), if \( BD \cdot BC = 5 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{5}}