inclusionAI/Ring-lite-rl-data huggingface dataset:
Analyze the recurrence $$T(n)=T\left(n-\frac{n}{\lg n}\right)+ O(n)$$ and determine its time complexity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(n \lg n)}
Five equilateral triangles of side length $2\sqrt{3}$ are arranged on the same side of a straight line, such that the midpoint of the base of each triangle is the vertex of the next triangle. Calculate the total area covered by these five triangles in the plane. Express your answer as a product of a number and $\sqrt{3}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12\sqrt{3}}
Calculate the sum of the sequence $2 - 6 + 10 - 14 + 18 - \cdots - 98 + 102$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-52}
Find the minimum value of the expression \( \frac{(1 + x + x^2)(1 + y + y^2)}{xy} \) given that \( x \neq 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
1. Petya came to the bus stop of the bus going to school with stops equally spaced from each other, and, not seeing the bus on the road, decided to run and get on the bus at the next stops along the way to school. Petya ran in such a way that at any moment he could notice the appearance of the bus on the road behind him. Seeing the bus, Petya can turn back or maintain his direction of movement. It is known that the speed of the bus is 5 times greater than Petya's running speed, and he can see the bus at a distance of no more than 0.6 km. Find the greatest value of the distance between stops, such that regardless of whether Petya turns back upon discovering the bus or not, he will be able to get on the bus at a stop. (the time the bus spends at the stop is not considered)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.25}
Determine the limiting distribution of \( h(X_n) = \sqrt{(\bar{X}_n - 1)^2} \) where \( X_1, \ldots, X_n, \ldots \) are Poisson random variables with parameter \( \theta \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|\theta - 1|}
Consider a 2D grid with \( n \) rows and \( m \) columns. You can fill this grid with numbers ranging from 1 to \( k \) (inclusive). The condition is that for each number \( r \) such that \( 1 \leq r \leq k \), no two rows can have exactly the same number of the number \( r \). Determine the number of ways to fill the grid under these conditions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Ten people are sitting around a round table. Each of them is either a knight, who always tells the truth, or a liar, who always lies. Each person was given a token. Then each of the people passed their token to one of their two neighbors. After this, 5 people said, “I have one token,” and the remaining 5 people said, “I have no tokens.” What is the maximum number of knights that could be among these 10 people?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
In an equilateral triangle ABC, three rods of lengths 3, 4, and 5 units are placed such that they intersect at a common point O, with the other ends at vertices A, B, and C, respectively. Given that BO = 3 units and CO = 4 units, find the angle ∠BOC.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150^\circ}
from an island, it is possible to reach the mainland by either ferry p, ferry q, or ferry r. Ferry p travels for 3 hours at 6 kilometers per hour, while ferry q takes a route that is three times longer than ferry p. If ferry p is slower than ferry q by 3 kilometers per hour, and ferry r takes a route that is 5 hours long and is half the speed of ferry q, find the difference in journey time between the slowest and fastest ferries.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Suppose  $ 2015= a_1 <a_2 < a_3<\cdots <a_k $  be a finite sequence of positive integers, and for all  $ m, n \in \mathbb{N} $  and  $1\le m,n \le k $ ,   $$  a_m+a_n\ge a_{m+n}+|m-n|  $$   Determine the largest possible value  $ k $  can obtain.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2016}
Find the largest integer $n \ge 3$ for which there exists an $n$-digit number $\overline{a_1a_2a_3\ldots a_n}$ with non-zero digits $a_1$, $a_2$, and $a_n$, such that the number is divisible by $\overline{a_2a_3\ldots a_n}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
A rope of length 10 *m* is tied tautly from the top of a flagpole to the ground 6 *m* away from the base of the pole. An ant crawls up the rope and its shadow moves at a rate of 30 *cm/min*. How many meters above the ground is the ant after 5 minutes? (This takes place on the summer solstice on the Tropic of Cancer so that the sun is directly overhead.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $S$ be the region enclosed by the curves $C_1: y = ax^2$ and $C_2: y = -ax^2 + 2abx$, where $a$ and $b$ are positive constants. Calculate the ratio of the volume $V_x$ of the solid formed by revolving $S$ around the $x$-axis to the volume $V_y$ of the solid formed by revolving $S$ around the $y$-axis. Find $\frac{V_x}{V_y}$. \( \frac{V_x}{V_y} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ab}
Let  $I$  be the set of points  $(x,y)$  in the Cartesian plane such that   $$ x>\left(\frac{y^4}{9}+2015\right)^{1/4} $$   Let  $f(r)$  denote the area of the intersection of  $I$  and the disk  $x^2+y^2\le r^2$  of radius  $r>0$  centered at the origin  $(0,0)$ . Determine the minimum possible real number  $L$  such that  $f(r)<Lr^2$  for all  $r>0$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{3}}
Calculate the number of elements of order 24 in the group \( C_{24} \times C_{40} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96}
For each natural number \( k \geqslant 2 \), a sequence \(\{a_{n}\}\) is defined by:
\[ a_{0} = k, \quad a_{n} = d(a_{n-1}), \quad n = 1, 2, 3, \cdots \]
where \( d(a) \) denotes the number of positive integer divisors of \( a \). Find all values of \( k \) such that the sequence \(\{a_{n}\}\) does not contain any perfect squares.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k \text{ is a prime number}}
The ratio of the two base angles in an isosceles triangle is 2:1. Find the measure of the vertex angle of this isosceles triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
Suppose that a polynomial of the form $p(x) = x^{2010} \pm x^{2009} \pm \cdots \pm x \pm 1$ has no real roots. What is the maximum possible number of coefficients of $-1$ in $p$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1005}
Let $ABCD$ be a cyclic quadrilateral with $AB = 5$, $BC = 10$, $CD = 11$, and $DA = 14$. The value of $AC + BD$ can be written as $\tfrac{n}{\sqrt{pq}}$, where $n$ is a positive integer and $p$ and $q$ are distinct primes. Find $n + p + q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{446}
Find, as a function of $\, n, \,$ the sum of the digits of \[9 \times 99 \times 9999 \times \cdots \times \left( 10^{2^n} - 1 \right),\] where each factor has twice as many digits as the previous one.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\[ 9 \cdot 2^n \]}
In a $7 \times 7$ grid, 19 squares are colored red. A row or column is considered to be red if it contains at least 4 red squares. What is the maximum number of red rows and columns in this grid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
A 2-kg rock is suspended by a massless string from one end of a uniform 1-meter measuring stick. The measuring stick is balanced by a support force at the 0.20-meter mark. Calculate the mass of the measuring stick. Express your answer in kg, rounded to two decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.33}
How many zeroes does the function $f(z) = e^z + z$ have in the square $[-10,10] \times [-6 \pi i, 6 \pi i]$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Given a stationary stochastic process with a spectral density \( S_{XX}(\omega) = 1 - \frac{|\omega|}{8 \pi} \), find the mean square value of the process.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
At the bake sale, Tamara makes $32 from the brownies. She made 2 pans of brownies which were all sold. The brownies were cut into a certain number of big square pieces, and each brownie cost $2. How many big square pieces were the brownies cut into?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
A classroom has  $30$  seats arranged into  $5$  rows of  $6$  seats. Thirty students of distinct heights come to class every day, each sitting in a random seat. The teacher stands in front of all the rows, and if any student seated in front of you (in the same column) is taller than you, then the teacher cannot notice that you are playing games on your phone. What is the expected number of students who can safely play games on their phone?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{163}{10}}
Two good trains, each 250 m long, are running in opposite directions on parallel tracks. One train has a certain speed, and the other train has a speed of 30 km/hr. The time taken by the slower train to pass the driver of the faster one is 23.998080153587715 seconds. What is the speed of the faster train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
On the sides of a right triangle, regular nonagons are constructed outside. These nonagons are built on one of the catheti and on the hypotenuse, with areas equal to $1602 \, \text{cm}^2$ and $2019 \, \text{cm}^2$, respectively. What is the area of the nonagon that is constructed on the other cathetus of this triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{417}
Given a table with numbers $1, 2, \ldots, 18$, determine the minimal number of numbers that must be erased so that the sum of any two remaining numbers is not a perfect square of a positive integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
78. The Walking Passenger. The train is moving at a speed of 60 km/h. A passenger is walking from the end of the train to its beginning through the passageways between the cars at a speed of 3 km/h. What is the speed of the passenger relative to the railway track?

We are not going to engage in sophistry here, like Zeno's paradox of the flying arrow, or Einstein's theory of relativity, but are talking about movement in the ordinary sense of the word relative to the railway track.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{63}
Find the diagonals of the quadrilateral formed by the angle bisectors of the internal angles of a rectangle with sides 1 and 3.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A \(3 \times 3\) square is partitioned into \(9\) unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated \(90\,^{\circ}\) clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. Find the probability that the grid is now entirely black. The original answer is in \(\frac{m}{n}\) format, please give the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{561}
Alice is planning a trip from the Bay Area to one of $5$ possible destinations, each serviced by only $1$ airport. She wants to book two flights: one to her destination and one returning. There are $3$ airports within the Bay Area from which she may leave and to which she may return. In how many ways can she plan her flight itinerary?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Given that the real numbers $a$, $b$, and $c$ satisfy $a \geq 1$, $b \geq 1$, $c \geq 1$, and $abc = 10$, and that $a^{\log_2{a}} \cdot b^{\log_2{b}} \cdot c^{\log_2{c}} \geq 10$, find the value of $a + b + c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Calculate the integral $\int_M(1-x_3)^2d\sigma$ over the closed upper hemisphere $M=\{(x_1,x_2,x_3) \in \mathbb{S}^2 | x_3 \ge 0\}$ in $\mathbb{R}^3$, where $d\sigma$ denotes the standard spherical measure. Can this be done without using spherical coordinates?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{2\pi}{3}$}
Let $H$ be a convex, equilateral heptagon whose angles measure (in degrees) $168^\circ$, $108^\circ$, $108^\circ$, $168^\circ$, $x^\circ$, $y^\circ$, and $z^\circ$ in clockwise order. Compute the number $y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{132}
Vladimir chose three digits \(a, b,\) and \(c\) such that \(a > b > c > 0\), and with them, he formed the numbers \(abc\), \(cba\), and \(cab\). Note that \(abc\) is not the product of \(a, b\), and \(c\), but rather the number composed of the digits \(a, b\), and \(c\). For example, if \(a = 1, b = 2\), and \(c = 3\), \(abc\) would be the number 123.

After choosing these three digits \(a, b\), and \(c\), Vladimir realized that one of the numbers formed was equal to the sum of the other two. Find the numbers formed by Vladimir.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{954, 459, 495}
平面上任意给定5个点，其中任三点不共线，则在以它们为顶点的三角形中，至多有＿个锐角三角形.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Let  $P(x), Q(x), $  and  $R(x)$  be three monic quadratic polynomials with only real roots, satisfying   $$ P(Q(x))=(x-1)(x-3)(x-5)(x-7) $$    $$ Q(R(x))=(x-2)(x-4)(x-6)(x-8) $$      for all real numbers  $x.$  What is  $P(0)+Q(0)+R(0)?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{129}
How many nonzero coefficients can a polynomial  $ P(x)$  have if its coefficients are integers and  $ |P(z)| \le 2$  for any complex number  $ z$  of unit length?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In the figure, there are two common tangents. These common tangents meet the circles at points \( A, B, C \), and \( D \). If \( AC = 9 \, \text{cm} \), \( BD = 3 \, \text{cm} \), \( \angle BAC = 60^\circ \), and \( AB = s \, \text{cm} \), find the value of \( s \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
当a为何值时，不等式$\log _ { \frac { 1 } { a } } ( \sqrt { x ^ { 2 } + a x + 5 } + 1 ) \cdot \log _ { 5 } ( x ^ { 2 } + a x +$ $6 ) + \log _ { a } 3 \geq 0$有且只有一个解？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the number of moles of HCl formed on combining 2 moles of C2H6 and 2 moles of Cl2. How many moles of HCl are formed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Each of the integers 1 to 7 is to be written, one in each circle in the diagram. The sum of the three integers in any straight line is to be the same. Find the number of different ways the centre circle can be filled. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Compute the $\textit{number}$ of ordered quadruples $(w,x,y,z)$ of complex numbers (not necessarily nonreal) such that the following system is satisfied:
\begin{align*}
wxyz &= 1\\
wxy^2 + wx^2z + w^2yz + xyz^2 &=2\\
wx^2y + w^2y^2 + w^2xz + xy^2z + x^2z^2 + ywz^2 &= -3 \\
w^2xy + x^2yz + wy^2z + wxz^2 &= -1\end{align*}
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Two squares, each with side length 5 cm, overlap as shown. The shape of their overlap is a square, which has an area of 4 cm². Find the perimeter, in centimeters, of the shaded figure. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Find the chromatic number of the graph obtained by removing $k$ vertex-disjoint edges from the complete bipartite graph $K_{m,n}$, where $k \leq \min\{m,n\}$. Express your answer in terms of $m$, $n$, and $k$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
The numbers $\sqrt{3v-2}$, $\sqrt{3v+1}$, and $2\sqrt{v}$ are the side lengths of a triangle. What is the measure of the largest angle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
Let $x,$ $y,$ and $z$ be real numbers such that $x + y + z = 7$ and $x, y, z \geq 2.$ Find the maximum value of
\[\sqrt{2x + 3} + \sqrt{2y + 3} + \sqrt{2z + 3}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{69}}
Find the largest real number \( k \) such that the inequality
\[
(a+b+c)^{3} \geq \frac{5}{2}\left(a^{3}+b^{3}+c^{3}\right)+k a b c
\]
holds for any side lengths \( a \), \( b \), and \( c \) of a triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{39}{2}}
Find the number of distinguishable groupings into which you can place $3$ indistinguishable red balls and $3$ indistinguishable blue balls. Here the groupings $RR-BR-B-B$ and $B-RB-B-RR$ are indistinguishable because the groupings are merely rearranged, but $RRB-BR-B$ is distinguishable from $RBB-BR-R$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
In the number \(2016 * * * * 02 *\), each of the 5 asterisks needs to be replaced by any of the digits \(0, 2, 4, 7, 8, 9\) (digits can repeat) so that the resulting 11-digit number is divisible by 6. In how many ways can this be done?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1728}
Let $\{a\_n\}$ be a geometric sequence with a common ratio of $q$ ($|q| > 1$). Define $b\_n = a\_n + 1 (n=1,2,...)$. If the sequence $\{b\_n\}$ has four consecutive terms in the set $\{-53, -23, 19, 37, 82\}$, find the value of $6q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-9}
Evaluate the limit: $$\lim_{n \to \infty}\displaystyle\sum_{r=1}^{n}\dfrac{r^3}{n^4 + n^2 + r^2}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/4}
Given that the line $l$ passing through $M(-2,0)$ intersects the ellipse $x^{2}+2y^{2}=2$ at points $P_{1}$ and $P_{2}$, and the midpoint of segment $P_{1}P_{2}$ is $P$. If the slope of line $l$ is $k_{1}$ ($k_{1}\neq 0$) and the slope of line $OP$ (where $O$ is the origin) is $k_{2}$, find the value of $k_{1}k_{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}
Using  $600$  cards,  $200$  of them having written the number  $5$ ,  $200$  having a  $2$ , and the other  $200$  having a  $1$ , a student wants to create groups of cards such that the sum of the card numbers in each group is  $9$ . What is the maximum amount of groups that the student may create?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Let $f(x) =x^4+14x^3+52x^2+56x+16.$  Let $z_1, z_2, z_3, z_4$ be the four roots of $f$.  Find the smallest possible value of $|z_az_b+z_cz_d|$ where $\{a,b,c,d\}=\{1,2,3,4\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given a number, we calculate its square and add $1$ to the sum of the digits in this square, obtaining a new number. If we start with the number $7$ we will obtain, in the first step, the number $1+(4+9)=14$, since $7^2 = 49$. What number will we obtain in the $1999$th step?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
11
}
9. Given that the increasing function $f(x)$ is defined on $(0,+\infty)$, and for any positive number $x$ it satisfies $f(x)$ $f\left[f(x)+\frac{1}{x}\right]=1$, find the value of $f(1)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1-\sqrt{5}}{2}}
Barney Schwinn notices that the odometer on his bicycle reads $1441$, a palindrome, because it reads the same forward and backward. After riding more hours that day and $6$ the next, he notices that the odometer shows another palindrome, $1661$. What was his average speed in miles per hour?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22
}
In acute $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. Given that $a+2b=4$ and $a\sin A+4b\sin B=6a\sin B\sin C$, find the minimum value of $c^{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5- \frac{4\sqrt{5}}{3}}
Let  $k$  and  $n$  be positive integers. Consider an array of  $2\left(2^n-1\right)$  rows by  $k$  columns. A  $2$ -coloring of the elements of the array is said to be *acceptable* if any two columns agree on less than  $2^n-1$  entries on the same row.

Given  $n$ , determine the maximum value of  $k$  for an acceptable  $2$ -coloring to exist.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ k = 2^n }
Determine the polynomials $p(x)$ for which the function $\sin p(x)$ is periodic.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p(x) = ax + b}
Solve the equation: $(2x+1)^2=3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{-1-\sqrt{3}}{2}}
Find all roots of the polynomial $x^4+2x^3-7x^2-8x+12$. Enter your answer as a list of numbers separated by commas.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1,2,-2,-3}
Find the minimal value of integer $ $ that guarantees: Among $ n$ sets, there exits at least three sets such that any of them does not include any other; or there exits at least three sets such that any two of them includes the other. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
5
}
Determine all polynomials \( P(x) \) with integer coefficients such that \( P(n) \) is a palindrome for every integer \( n \geq 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{P(x) = c}
A box $2$ centimeters high, $3$ centimeters wide, and $5$ centimeters long can hold $40$ grams of clay. A second box with twice the height, three times the width, and the same length as the first box can hold $n$ grams of clay. What is $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$240$}
Given that $F_1$ and $F_2$ are the left and right foci of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), and $P$ is a point on the ellipse, with $\overrightarrow{PF_{1}} \cdot (\overrightarrow{OF_{1}} + \overrightarrow{OP}) = 0$, if $|\overrightarrow{PF_{1}}| = \sqrt{2}|\overrightarrow{PF_{2}}|$, determine the eccentricity of the ellipse.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{6} - \sqrt{3}}
The area of the largest square that can be inscribed in a regular hexagon with side length $1$ can be expressed as $a - b\sqrt{c}$, where $c$ is not divisible by the square of any prime. Find $a + b + c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}