inclusionAI/Ring-lite-rl-data huggingface dataset:
$[\quad$ Evaluation + example $\quad]$

The hostess baked a pie for her guests. She may have either 10 or 11 guests. Into what smallest number of slices should she cut the pie in advance so that it can be evenly divided among either 10 or 11 guests?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Find all pairs \((n, p)\) of natural numbers \(n\) and prime numbers \(p\) that satisfy the equation \(p(p-1) = 2(n^3 + 1)\). Calculate the sum \(\sum (n_i + p_i)\) for all such pairs \((n_i, p_i)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{147}
Determine the smallest positive integer that has 144 positive divisors, and among these divisors, there are ten consecutive ones.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{110880}
设$\frac { 1 } { 1 + \frac { 1 } { 1 + \frac { 1 } { 1 + \cdots } + \frac { 1 } { 1 } } } = \frac { m } { n } $,

其中$m$和$n$是互质的自然数，而等式左边含有1988条分数线，试计算$m ^ { 2 } + m n - n ^ { 2 }$的值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Let $\mathcal{T}$ be the set of all linear maps $A$ from $\mathbb{R}^2$ such that $A^4 - 5A^2 + 4I = 0$. Determine the maximum number of elements in $\mathcal{T}$ such that no two maps are similar.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Consider a sequence formed from the numbers $1, 2, 3, 4, 5, 6, 7$, where all $7!$ permutations are equally likely. What is the probability that there will be at least five consecutive positions in which the numbers are in increasing order?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{45}}
Evaluate the limit: $$\lim_{\epsilon \to 0} \int_\mathbb {R} \frac{1}{x^2-(1-\ic\epsilon)^2} \, dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-i\pi}
Given independent exponential random variables $X_1, X_2, X_3, X_4$ with rates 1, 2, 3, and 4, respectively, calculate the probability $P(X_1<X_2<X_3<X_4|X_2<X_4)$. Use the formula for conditional probability to find your answer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{35}}
A particle moves along a straight line with a position function $x(t)$. At any instant, the product of its position and acceleration equals the square of its velocity. Given that the particle starts at position $x(0) = 1$ with initial velocity $v(0) = 0.1$, find the position function $x(t)$. Provide your answer as a function of $t$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{0.1t}}
Two teams, Team A and Team B, each consist of 10 people. Team A selects and removes n people from its team, where 1 ≤ n ≤ 9. Team B then selects m people from its team to replace with m of the n people removed from Team A, where 0 ≤ m ≤ n. Determine the number of distinct possible combinations of teams after these changes, and find the remainder when this number is divided by 1000.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{696}
Determine the maximum number of intersections between the polynomials \( g(x) = a_1x + a_2x^2 + \ldots + a_kx^k \) and \( f(x) = b_jx^j \), where \( a_1, a_2, \ldots, a_k > 0 \), \( j \in \{1, 2, \ldots, k-1\} \), \( b_j > 0 \), and \( x \geq 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let $U$ be an open set of the Riemann sphere, and let $z_i$ be $n$ distinct points in $U$. Consider $E$, the vector space of meromorphic functions on $U$ with poles of order no more than 2, and $F$, the subspace of $E$ consisting of functions that are holomorphic in a neighborhood of each $z_i$. Determine if the quotient space $E/F$ has finite dimension, and if so, find its dimension.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n}
Calculate the variance of the expression \(12(X_1 + X_2 + X_3)\), where \(X_1\) is a random variable uniformly distributed on \((0,1)\), \(X_2\) is uniformly distributed on \((0,X_1)\), and \(X_3\) is uniformly distributed on \((0,X_2)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{565}{12}}
Determine the eigenvalues of the operator $D^2$ for which the eigenfunctions are 2π-periodic.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-n^2}
Four siblings ordered an extra large pizza. Alex ate \( \frac{1}{5} \) ,Beth \( \frac{1}{3} \) ,and Cyril \( \frac{1}{4} \) of the pizza. Dan got the leftovers. What is the
 sequence of the siblings in decreasing order of the part of pizza they consumed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
Beth, Cyril, Dan, Alex
}
In a qualification group with  $15$  volleyball teams, each team plays with all the other teams exactly once. Since there is no tie in volleyball, there is a winner in every match. After all matches played, a team would be qualified if its total number of losses is not exceeding  $N$ . If there are at least  $7$  teams qualified, find the possible least value of  $N$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
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 transformations $T_1, T_2, T_3, \dots, T_n$ returns the point $(1,0)$ back to itself?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
359
}
Find the volume of the region in the first octant (where $x \ge 0$, $y \ge 0$, $z \ge 0$) bounded by the cylinder $x^2 + y^2 = 4$ and the planes $z = 2$ and $y + z = 4$. Evaluate the integral \( \int_0^2 \int_0^{4-z} \int_0^{\sqrt{4-y^2}} dx \, dy \, dz \) to find the volume.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Let \( z \) be a complex number satisfying \( \left( {z + \frac{1}{z}}\right) \left( {z + \frac{1}{z} + 1}\right)  = 1 \) . Evaluate
 \( \left( {3{z}^{100} + \frac{2}{{z}^{100}} + 1}\right) \left( {{z}^{100} + \frac{2}{{z}^{100}} + 3}\right) \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Count the number of permutations in $S_5$ that are a product of exactly two disjoint cycles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Given a cyclic quadrilateral \(ABCD\), it is known that \(\angle ADB = 48^{\circ}\) and \(\angle BDC = 56^{\circ}\). Inside the triangle \(ABC\), a point \(X\) is marked such that \(\angle BCX = 24^{\circ}\), and the ray \(AX\) is the angle bisector of \(\angle BAC\). Find the angle \(CBX\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{38^\circ}
Evaluate the following limit:

$$\lim \limits_{x \to 0} \frac{(\sin{x})^{x-1}}{\frac{1}{x}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $ABCD$ be a rectangle with $AD = 1$. Let $P$ be a point on diagonal $\overline{AC}$, and let $\omega$ and $\xi$ be the circumcircles of $\vartriangle APB$ and $\vartriangle  CPD$, respectively. Line $\overleftrightarrow{AD}$ is extended, intersecting $\omega$ at $X$, and $\xi$ at $Y$ . If $AX = 5$ and $DY = 2$, find $[ABCD]^2$.
Note: $[ABCD]$ denotes the area of the polygon $ABCD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Cars A and B start from points A and B simultaneously and move towards each other. Car A travels at a higher speed than Car B. After Car A has traveled 60 kilometers, they meet at point C. Car A continues forward while Car B turns around and follows in the same direction as Car A. Calculate the maximum distance between the two cars when Car A reaches point B. Express your answer in kilometers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Suppose the state of Georgia uses a license plate format "LLDLLL", and the state of Nebraska uses a format "LLDDDDD". Assuming all 10 digits are equally likely to appear in the numeric positions, and all 26 letters are equally likely to appear in the alpha positions, how many more license plates can Nebraska issue than Georgia?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21902400}
$M$ is the midpoint of the side $AB$ in an equilateral triangle $\triangle ABC$. The point $D$ on the side $BC$ is such that $BD : DC = 3 : 1$. On the line passing through $C$ and parallel to $MD$, there is a point $T$ inside the triangle $\triangle ABC$ such that $\angle CTA = 150^\circ$. Find the $\angle MTD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
The set \( A \) consists of integers, with the smallest element being 1 and the largest being 100. Except for 1, every element is equal to the sum of two (not necessarily distinct) elements from the set \( A \). Find the minimum number of elements in set \( A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Find the volume of the region of points $(x,y,z)$ such that $$(x^2+y^2+z^2+8)^2\leq36(x^2+y^2).$$ The original answer is in the format $m\pi^{n}$, please give the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
A man can row his boat with the stream at 22 km/h and against the stream at some speed. The man's rate is 6 km/h. What is his speed against the stream?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Given that in triangle $\triangle ABC$, $a=2$, $\angle A=\frac{π}{6}$, $b=2\sqrt{3}$, find the measure of angle $\angle C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{2}}
A city has a section of road that needs to be repaired, which is no longer than 3500 meters. There are three construction teams, A, B, and C, working on the sidewalk, the non-motorized lane, and the motorized lane, respectively. They start working at the same time at midnight and work continuously for 24 hours a day. Several days later at midnight, team A completes its task; a few days later at 18:00, team B completes its task, and starting from midnight on the day team B completes its task, a few days later at 08:00, team C completes its task. It is known that the daily completion rates of the three construction teams are 300 meters, 240 meters, and 180 meters, respectively. How long is this section of the road?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3300}
Two individuals, A and B, participate in an exam that consists of a pool of 10 questions. A can answer 6 of these questions correctly, while B can answer 8 correctly. For each test, 3 questions are randomly selected from the pool. The passing criterion is to answer at least 2 questions correctly. Calculate the probability that at least one of A or B passes the test. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{44}{45}}
The area of a circle is given by $A = \pi r^2$. If the area is $A = \pi(3m^2 + 30m + 75)$, what is the radius of the circle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}(m+5)}
The functions \( f(x) \) and \( g(x) \) are defined for all \( x > 0 \). The function \( f(x) \) is equal to the greater of the numbers \( x \) and \( \frac{1}{x} \), and \( g(x) \) is equal to the smaller of the numbers \( x \) and \( \frac{1}{x} \). Solve the equation \( f(5x) \cdot g(8x) \cdot g(25x)=1 \). In the answer, specify the solution if it is unique, or the sum of the solutions if there are several. Round the answer to the nearest hundredth if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.09}
The constant term in the expansion of (1+x)(e^(-2x)-e^x)^9.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
Find the number of lines that pass through the point $(0,1)$ and intersect the parabola $y^2=4x$ at only one point.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Steve wrote the digits $1$, $2$, $3$, $4$, and $5$ in order repeatedly from left to right, forming a list of $10,000$ digits, beginning $123451234512\ldots.$ He then erased every third digit from his list (that is, the $3$rd, $6$th, $9$th, $\ldots$ digits from the left), then erased every fourth digit from the resulting list (that is, the $4$th, $8$th, $12$th, $\ldots$ digits from the left in what remained), and then erased every fifth digit from what remained at that point. What is the sum of the three digits that were then in the positions $2019, 2020, 2021$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
For any integer $k \geq 1$, let $p(k)$ be the smallest prime which does not divide $k.$ Define the integer function $X(k)$ to be the product of all primes less than $p(k)$ if $p(k) > 2$, and $X(k) = 1$ if $p(k) = 2.$ Let $\{x_n\}$ be the sequence defined by $x_0 = 1$, and $x_{n+1}X(x_n) = x_n p(x_n)$ for $n \geq 0.$ Find the smallest positive integer $t$ such that $x_t = 2090.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{149}
For the system of equations $|x_i-x_j|=\alpha$ with $x_i \in \mathbb{R}^n$, what is the minimum number of equations that can be satisfied for a general $n$, assuming that the $x_i$ values cannot be all equal?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In the acute triangle ABC, the sides opposite to the angles A, B, and C are a, b, and c respectively, with a > b > c. If the side PQ of the inscribed square PQRS falls on side a of triangle ABC, and the other two vertices R and S lie on sides b and c respectively, let the side length of square PQRS be x_a. Similarly, define x_b and x_c. Determine the correct relationship among x_a, x_b, and x_c. Express your answer using inequality symbols.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x_a < x_b < x_c}
Maria ordered a certain number of televisions at $R$ \$ 1994.00 each. She noticed that in the total amount to be paid, there are no digits 0, 7, 8, or 9. What was the smallest number of televisions she ordered?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
There are  $n\geq 3$  pupils standing in a circle, and always facing the teacher that stands at the centre of the circle. Each time the teacher whistles, two arbitrary pupils that stand next to each other switch their seats, while the others stands still. Find the least number  $M$  such that after  $M$  times of whistling, by appropriate switchings, the pupils stand in such a way that any two pupils, initially standing beside each other, will finally also stand beside each other; call these two pupils  $ A$  and  $ B$ , and if  $ A$  initially stands on the left side of  $ B$  then  $ A$  will finally stand on the right side of  $ B$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{(n-1)n}{2}}
Evaluate the integral \( I = \int_0^1 \log \Gamma(x) \cos (2\pi n x)\, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4n}}
In a factory, the average monthly salary of laborers and supervisors is Rs. 1250 per month. There are 6 supervisors with an average monthly salary of Rs. 2450. There are 42 laborers. What is the average monthly salary of the laborers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1078.57}
In the convex quadrilateral $ABCD$, it is known that $\angle CBD = 58^\circ$, $\angle ABD = 44^\circ$, and $\angle ADC = 78^\circ$. Find the angle $\angle CAD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{58^\circ}
Evaluate the limit: $$\lim_{n\to\infty}\,\frac {n\left( 3\,{{\rm e}^{3}} \left( \frac {n+1}{n} \right) ^{-3\,n}n-1-3\,n \right)}{n+1} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{2}}
The perimeter of an isosceles triangle is 24. A median to one of the equal sides divides the perimeter into two parts in the ratio 5:3. Find the length of the base of this triangle. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Evaluate the integral \( \int_{\mathbb{R}} (F(x+b)-F(x+a)) \, dx \) for any real numbers \(a\) and \(b\), where \(F\) is a distribution function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{b - a}
Determine the following limit: $$\lim\limits_{x\to0}\frac{\left(1+x\cdot e^x\right)^{\frac{1}{x^2}}}{\left(1+x\cdot5^x\right)^{\frac{1}{x^2}}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{e}{5}}
For any real numbers sequence  $\{x_n\}$  ,suppose that  $\{y_n\}$  is a sequence such that: $y_1=x_1, y_{n+1}=x_{n+1}-(\sum\limits_{i = 1}^{n} {x^2_i})^{ \frac{1}{2}}$   ${(n \ge 1})$   .
Find the smallest positive number   $\lambda$  such that  for any real numbers sequence  $\{x_n\}$  and all positive integers  $m$  , have  $\frac{1}{m}\sum\limits_{i = 1}^{m} {x^2_i}\le\sum\limits_{i = 1}^{m} {\lambda^{m-i}y^2_i} .$ (High School Affiliated to Nanjing Normal University )
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\lambda = 2}
How many ordered triples \((a, b, c)\) of positive integers are there such that none of \(a, b, c\) exceeds 2010 and each of \(a, b, c\) divides \(a + b + c\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9045}
Define  $\phi_n(x)$  to be the number of integers  $y$  less than or equal to  $n$  such that  $\gcd(x,y)=1$ . Also, define  $m=\text{lcm}(2016,6102)$ . Compute  $$ \frac{\phi_{m^m}(2016)}{\phi_{m^m}(6102)}. $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{339}{392}}
How many distinct arrangements are possible for wearing five different rings in the five fingers of the right hand? (We can wear multiple rings in one finger)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$5^5$

}
Given the equation \(1+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{23} = \frac{a}{23!}\), find the remainder when \(a\) is divided by 13.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Determine the number of distinct arrangements of the letters in the word "MATHEMATICS" such that no two vowels are adjacent and the letter 'T' appears before the letter 'S'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{352800}
Compute the derivative \( f'(e^{i 2\pi/3}) \) for the function \( f(z) = z^{1/4} \), where the branch is defined in \( \mathbb{C} - (\infty,0] \) and satisfies \( f(1) = -i \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{4}}
Consider the set of points \(S\) where each point \((x, y)\) has integer coordinates with \(1 \leq x, y \leq 26\). Determine the number of ways to select a convex subset of \(S\) that contains exactly 98 points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{520}
Nathan is buying decorations for his wedding reception. The reception hall will have 30 tables. Each table needs a linen tablecloth ($25 to rent), 6 place settings ($12 each to rent), and a centerpiece. Each centerpiece will have 15 roses ($6 each, but there is a 10% discount for bulk purchase), 20 lilies ($5 each, but there is a 5% discount for bulk purchase), 5 daisies ($3 each), and 3 sunflowers ($4 each). Additionally, Nathan needs to purchase lighting for the hall at a cost of $450, and tablecloth rentals come with a 15% service charge. How much will the decorations cost after all the discounts and additional charges are applied?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$9562.50}
Gary is baking cakes, one at a time. However, Gary’s not been having much success, and each failed cake will cause him to slowly lose his patience, until eventually he gives up. Initially, a failed cake has a probability of $0$ of making him give up. Each cake has a $1/2$ of turning out well, with each cake independent of every other cake. If two consecutive cakes turn out well, the probability resets to $0$ immediately after the second cake. On the other hand, if the cake fails, assuming that he doesn’t give up at this cake, his probability of breaking on the next failed cake goes from p to $p + 0.5$. If the expected number of successful cakes Gary will bake until he gives up is$ p/q$, for relatively prime $p, q$, find $p + q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{86}
Find the order of $257$ in $\Bbb Z/660$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Given an arithmetic sequence $\{a_n\}$, it is known that $\frac{a_{11}}{a_{10}} + 1 < 0$. Determine the maximum value of $n$ for which $S_n > 0$ holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
In a six-digit decimal number $\overline{a_{1} a_{2} a_{3} a_{4} a_{5} a_{6}}$, each digit $a_{i}(1 \leqslant i \leqslant 6)$ is an odd number, and the digit 1 is not allowed to appear consecutively (for example, 135131 and 577797 satisfy the conditions, while 311533 does not satisfy the conditions). Find the total number of such six-digit numbers. $\qquad$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13056}
Three people, A, B, and C, are standing in a row from left to right. If two people are randomly selected for a position swap, what is the probability that A will be to the left of B after two such swaps?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{3}}
Find the quotient group \( \frac{\mathbb{Z}\times\mathbb{Z}\times\mathbb{Z}}{\langle(1,1,1),(1,3,2)\rangle} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}}
Evaluate the double integral \( \iint_{D}\left(54 x^{2} y^{2}+150 x^{4} y^{4}\right) d x d y \) where the region \(D\) is bounded by the lines \(x=1, y=x^{3}\), and \(y=-\sqrt{x}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Amy, Bomani, Charlie, and Daria work in a chocolate factory. On Monday Amy, Bomani, and Charlie started working at $1{:}00 \text{ PM}$ and were able to pack $4$, $3$, and $3$ packages, respectively, every $3$ minutes. At some later time, Daria joined the group, and Daria was able to pack $5$ packages every $4$ minutes. Together, they finished packing $450$ packages at exactly $2{:}45 \text{ PM}$. At what time did Daria join the group?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$1{:}25\text{ PM}$
}
Calculate the limit of the numerical sequence:

$$
\lim _{n \rightarrow \infty}\left(\frac{3 n^{2}-6 n+7}{3 n^{2}+20 n-1}\right)^{-n+1}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{\frac{26}{3}}}
Beth, Carolyn, and George take turns reading a 20-page story in the order Beth, Carolyn, then George. One evening, Beth is absent, and Carolyn and George read the story in turns starting with Carolyn. How many pages are read by the same person in both the original and modified reading sequences? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}