inclusionAI/Ring-lite-rl-data huggingface dataset:
Let $X$ be the number of dice tosses required to obtain the first even number, and $Y$ the number of tosses for the first number greater than 3. Both $X$ and $Y$ are geometric random variables with a mean of 2. Determine the probability $P(X=Y)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{1}{2}$}
It is desired to construct a right triangle in the coordinate plane so that its legs are parallel to the $x$ and $y$ axes and so that the medians to the midpoints of the legs lie on the lines $y = 3x + 1$ and $y = mx + 2$. The number of different constants $m$ for which such a triangle exists is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2$}
For a sequence $a_{1}<a_{2}<\cdots<a_{n}$ of integers, a pair ( $a_{i}, a_{j}$ ) with $1 \leq i<$ $j \leq n$ is called \underline{interesting} if there exists a pair ( $a_{k}, a_{l}$ ) of integers with $1 \leq k<l \leq n$ such that  
$$\frac{a_{l}-a_{k}}{a_{j}-a_{i}}=2$$  
For each $n \geq 3$, find the largest possible number of interesting pairs in a sequence of length $n$. Please provide the value of this expression when $n = 5$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
A cowboy is 4 miles south of a stream which flows due east. He is also 8 miles west and 7 miles north of his cabin. He wishes to water his horse at the stream and return home. The shortest distance (in miles) he can travel and accomplish this is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$17$}
设函数 f(z)（z 为复数）满足 f(f(z))=(z\overline{z}-z-\overline{z})^2，若 f(1)=0，则 |f(i)-1|=__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the greatest constant $C$ so that for all real numbers $x_1, x_2,\ldots,x_6$ the inequality $$(x_1 + x_2 + ...+ x_6)^2 \geq C \cdot (x_1 (x_2 + x_3) + x_2 (x_3 + x_4) + \ldots+ x_6 (x_1 + x_2))$$ applies.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given a linear function $y=x-k$, if for any value of the independent variable $x$ in the range $x \lt 3$, the corresponding function value $y$ is less than $2k$, then the range of values for $k$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k \geq 1}
Between a certain natural number and 81, there are 15 consecutive natural numbers. What is this certain number if it is a natural number less than 81?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65}
Reggie is playing marbles with his friend. His friend arrives with some marbles. Each game, they bet ten marbles and whoever wins gets to keep all the marbles. After 9 games, Reggie has 90 marbles and lost 1 game. How many marbles did his friend arrive with?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
the regular price per can of a certain brand of soda is $ 0.40 . if the regular price per can is discounted 15 percent when the soda is purchased in 24 - can cases , what is the price of 100 cans of this brand of soda purchased in 24 - can cases ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$34.00}
Given that $c > 0$, suppose proposition $p$: the function $y=(2c-1)^{x}$ is monotonically decreasing on $\mathbb{R}$; proposition $q$: the solution set of the inequality $x+|x-2c| > 1$ is $\mathbb{R}$. If $p \vee q$ is true and $p \wedge q$ is false, find the range of values for $c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{c\geqslant 1}
Five runners ran a relay race. If the first runner had run twice as fast, they would have spent 5% less time. If the second runner had run twice as fast, they would have spent 10% less time. If the third runner had run twice as fast, they would have spent 12% less time. If the fourth runner had run twice as fast, they would have spent 15% less time. By what percentage less time would they have spent if the fifth runner had run twice as fast?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8\%}
Determine the value of the integral $\int_0^{\infty} \frac{dx}{(1+x^n)(1+x^2)}$ for any $n \neq 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{\pi}{4}$}
Round up the smallest six-digit number that you can make by using the six number cards 0, 1, 3, 5, 7, and 9 once, to the thousandth place.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{104000}
In unit square \(ABCD\), points \(E, F, G\) are chosen on sides \(BC, CD,\) and \(DA\) respectively such that \(AE\) is perpendicular to \(EF\) and \(EF\) is perpendicular to \(FG\). Given that \(GA = \frac{404}{1331}\), find all possible values of the length of \(BE\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9}{11}}
Find the limit of the function $$ \lim_{x\rightarrow 0}\left( \frac{1}{x\cos(x)\tan(x\cos(x))}-\frac{1}{x^2\cos^2(x)} \right) $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{3}}
A circle centered at $O$ has radius 1 and contains the point $A$. Segment $AB$ is tangent to the circle at $A$ and $\angle
AOB=\theta$. If point $C$ lies on $\overline{OA}$ and $\overline{BC}$ bisects $\angle ABO$, then express $OC$ in terms of $s$ and $c,$ where $s = \sin \theta$ and $c = \cos \theta.$

[asy]
pair A,B,C,O;
O=(0,0);
A=(1,0);
C=(0.6,0);
B=(1,2);
label("$\theta$",(0.1,0),NE);
label("$O$",O,S);
label("$C$",C,S);
label("$A$",A,E);
label("$B$",B,E);
draw(A--O--B--cycle,linewidth(0.7));
draw(C--B,linewidth(0.7));
draw(Circle(O,1),linewidth(0.7));
[/asy]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{1+s}}
Calculate the limit: $$ \lim_{n\to\infty} \frac{2^n + \left(1+\frac{1}{n^2}\right)^{n^3} + \frac{4^n}{n^4}}{\frac{4^n}{n^4} + n^3\cdot 3^n} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the number of distinct necklaces that can be formed using 15 identical beads and 3 different gems, taking into account rotations and reflections.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{91}
Task 1. We have 1000 balls in 40 different colors, with exactly 25 balls of each color. Determine the smallest value of $n$ with the following property: if you randomly arrange the 1000 balls in a circle, there will always be $n$ balls next to each other where at least 20 different colors occur.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{452}
Given the sequence $\{a\_n\}$ that satisfies: $(a\_1=-13$, $a\_6+a\_8=-2)$, and $(a_{n-1}=2a\_n-a_{n+1}\ (n\geqslant 2))$, find the sum of the first 13 terms of the sequence $\left\{\frac{1}{a\_n a_{n+1}}\right\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{13}}
Households A, B, and C plan to subscribe to newspapers. There are 5 different types of newspapers available. Each household subscribes to two different newspapers. It is known that each pair of households shares exactly one common newspaper. How many different subscription ways are there for the three households?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{180}
Given the real numbers \( x_1, x_2, \ldots, x_{2001} \) satisfy \( \sum_{k=1}^{2000} \left|x_k - x_{k+1}\right| = 2001 \). Let \( y_k = \frac{1}{k} \left( x_1 + x_2 + \cdots + x_k \right) \) for \( k = 1, 2, \ldots, 2001 \). Find the maximum possible value of \( \sum_{k=1}^{2000} \left| y_k - y_{k+1} \right| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2000}
Find the area of the region defined by the inequality: \( |y - |x - 2| + |x|| \leq 4 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Evaluate $$\lim_{x \to 0}\frac{\tan(\tan x)-\sin(\sin x)}{\tan x-\sin x}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let \( S \) be the set of all non-negative integers with at most \( n \) digits in decimal, and \( S_{k} \) be the subset of \( S \) consisting of elements whose digit sum is less than \( k \). Determine the value of \( n \) for which there exists a \( k \) such that \( |S| = 2|S_{k}| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $N = 6 + 66 + 666 + \ldots + \underbrace{666\ldots66}_{100\text{ sixes}}$, where there are one hundred $6$'s in the last term in the sum. How many times does the digit $7$ occur in the number $N$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
Allie has 18 toys in total worth $108. There are 3 types of toys: Type A, Type B, and Type C. We know that one Type A toy is worth $14, and there are 4 Type A toys. Moreover, there are 7 Type B toys and they all have the same value. Finally, there are 7 Type C toys and they also have the same value, but it is different than the value of Type B toys. If we know that the combined value of all Type B and Type C toys is $84, how much does one Type B toy cost?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$6}
Let point \( P \) be on the ellipse \( \frac{x^{2}}{5}+y^{2}=1 \), and let \( F_{1} \) and \( F_{2} \) be the two foci of the ellipse. If the area of \( \triangle F_{1} P F_{2} \) is \( \frac{\sqrt{3}}{3} \), find \( \angle F_{1} P F_{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^
\circ}
Six different digits from the set  
$\{ 1,2,3,4,5,6,7,8,9\}$ 
are placed in the squares in the figure shown so that the sum of the entries in the vertical column is $23$ and the sum of the entries in the horizontal row is $12$.  
What is the sum of the six digits used?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
定义“$\frac{{{\left| x \right|}^{n}}}{{{an}}}+\frac{{{\left| y \right|}^{n}}}{{{bn}}}=1\left( a>b>0,n\in {{N}^{*}},n\ge 3 \right)$”代表的曲线为“超椭圆”，设$a$、$b$为常数，设超椭圆的周长为${{C}_{n}}$，那么$\underset{n\to \infty }{\mathop{\lim }}\,{{Cn}}=$______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$4a+4b$}
On a grid sheet, a $1 \times 5$ rectangle is initially shaded. Each minute, all unshaded cells that have at least one shaded neighboring cell are shaded. For example, after one minute, 17 cells will be shaded. How many cells will be shaded after 5 minutes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{105}
The sphere with radius 1 and center $(0,0,1)$ rests on the $xy$-plane.  A light source is at $P = (0,-1,2).$  Then the boundary of the shadow of the sphere can be expressed in the form $y = f(x),$ for some function $f(x).$  Find the function $f(x).$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{x^2}{4} - 1}
The 5th root of what integer power of 8 is closest to 100?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Given that $\angle BAC = 90^{\circ}$ and the quadrilateral $ADEF$ is a square with side length 1, find the maximum value of $\frac{1}{AB} + \frac{1}{BC} + \frac{1}{CA}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 + \frac{\sqrt{2}}{2}}
Divide the plane into 1000 regions using the fewest possible number of lines.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
In triangle $ABC$, $AB = 100$, $BC = 120$, and $CA = 140$. Points $D$ and $F$ lie on $\overline{BC}$ and $\overline{AB}$, respectively, such that $BD = 90$ and $AF = 60$. Point $E$ is an arbitrary point on $\overline{AC}$. Denote the intersection of $\overline{BE}$ and $\overline{CF}$ as $K$, the intersection of $\overline{AD}$ and $\overline{CF}$ as $L$, and the intersection of $\overline{AD}$ and $\overline{BE}$ as $M$. If $[KLM] = [AME] + [BKF] + [CLD]$, where $[X]$ denotes the area of region $X$, compute $CE$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{91}
Two types of pieces, bishops and rooks, are to be placed on a $10 \times 10$ chessboard (without necessarily filling it) such that each piece occupies exactly one square of the board. A bishop $B$ is said to attack a piece $P$ if $B$ and $P$ are on the same diagonal and there are no pieces between $B$ and $P$ on that diagonal; a rook $R$ is said to attack a piece $P$ if $R$ and $P$ are on the same row or column and there are no pieces between $R$ and $P$ on that row or column. A piece $P$ is chocolate if no other piece $Q$ attacks $P$. What is the maximum number of chocolate pieces there may be, after placing some pieces on the chessboard?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
A ladder is formed by removing some consecutive unit squares of a $10 \times 10$ chessboard such that for each $k$-th row ($k \in \{1,2,\dots, 10\}$), the leftmost $k-1$ unit squares are removed. Find the number of rectangles formed by the composition of unit squares that the ladder has.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{715}
Consider the sum:
\[
  S_n = \sum_{k = 1}^n \frac{1}{\sqrt{2k-1}} 
\]
Determine \( \lfloor S_{4901} \rfloor \). Recall that if \( x \) is a real number, then \( \lfloor x \rfloor \) (the floor of \( x \)) is the greatest integer that is less than or equal to \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{98}
已知集合 $A=\left\{x \mid \log _{a}(a x-1)>1\right\}$ 。若 $3 \in A$, 请给出 a 的取值范围中所有有限端点的和的6倍。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
In the convex quadrilateral \(ABCD\), the midpoint of side \(AD\) is marked as point \(M\). Segments \(BM\) and \(AC\) intersect at point \(O\). It is known that \(\angle ABM = 55^\circ\), \(\angle AMB = 70^\circ\), \(\angle BOC = 80^\circ\), and \(\angle ADC = 60^\circ\). How many degrees is the angle \(BCA\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35^\circ}
A person distributed 20% of his income to his 3 children each. He deposited 30% of his income to his wife's account. He donated 5% of the remaining amount to an orphan house. Finally, he has $40,000. What is his total income?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$421,052.63}
A total of 2021 coloured koalas are arranged in a row and are numbered from 1 to 2021. Each koala is coloured red, white, or blue. Amongst any three consecutive koalas, there are always koalas of all three colours. Sheila guesses the colours of five koalas: Koala 2 is white; Koala 20 is blue; Koala 202 is red; Koala 1002 is blue; Koala 2021 is white. Only one of her guesses is wrong. Find the number of the koala whose colour she guessed incorrectly. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
The sum of the two base angles of a trapezoid is 90 degrees. The length of the upper base is 5, and the length of the lower base is 11. Find the length of the segment connecting the midpoints of the two bases. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let  $M$  be a set of  $99$  different rays with a common end point in a plane. It's known that two of those rays form an obtuse angle, which has no other rays of  $M$  inside in. What is the maximum number of obtuse angles formed by two rays in  $M$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3267}
Let \( M = \{1, 2, 3, \ldots, 40\} \). Find the smallest positive integer \( n \) such that \( M \) can be partitioned into \( n \) pairwise disjoint subsets, and for any 3 numbers \( a, b, c \) (not necessarily distinct) within the same subset, \( a \neq b + c \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Una rolls $6$ standard $6$-sided dice simultaneously and calculates the product of the $6$ numbers obtained. What is the probability that the product is divisible by $4$? Provide your answer as a simplified fraction in the form \frac{k}{m}. Please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{123}
求最小正整数 $n$, 使得任取 $n$ 个不同的正整数中必存在 8 个不同的正整数,其和被 8整除
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
My grandson is about as many days as my son in weeks, and my grandson is as many months as I am in years. My grandson, my son, and I together are a certain number of years. If I am 72 years old, how many years are we together?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Given a sequence of 99 terms \( (p_1, p_2, \cdots, p_{99}) \) with a Cesaro sum of 1000, find the Cesaro sum of a new 100-term sequence \( (1, p_1, p_2, \cdots, p_{99}) \). Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{991}
Given that the incircle of quadrilateral ABCD touches sides AB, BC, CD, and DA at points E, F, G, and H respectively, and AF, DF, and EG intersect at points M and N. If BF = CF = 5 and EG = 6, then find the length of MN. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
In assigning dormitory rooms, a college gives preference to pairs of students in this order: $$ A A, A B, A C, B B, B C, A D, C C, B D, C D, D D $$ in which $A A$ means two seniors, $A B$ means a senior and a junior, etc. Determine numerical values to assign to $A, B, C, D$ so that the set of numbers $A + A, A + B, A + C, B + B$, etc., corresponding to the order above will be in descending magnitude. Please provide the sum of A + B + C + D for the least positive integers that satisfy this condition.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
Modern widescreen television screens are rectangles and are typically measured by the length of their diagonals. The ratio of the horizontal length to the height in these widescreen televisions is $16:9$. Calculate the horizontal length (in inches) of a ``50-inch'' widescreen television screen.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43.6}
A rectangular grazing area is to be fenced off on three sides using part of a 100 meter rock wall as the fourth side. Fence posts are to be placed every 12 meters along the fence, including the two posts where the fence meets the rock wall. What is the minimum number of fence posts required to fence an area 36 m by 60 m? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Find the maximum number of $1$'s in a square matrix of size $n$ such that the matrix remains invertible, given that all the entries are either $0$ or $1$ and not all the diagonal entries are $0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^2-n+1}
Given two power series, where one converges for \(|x| < 6\) and the other converges for \(|x| < 4\), determine the interval of convergence for the product of these two series.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-4, 4)}
Evaluate the limit: \[ \lim_{n\to\infty} \sum_{r=0}^n \frac{\binom{n}{r}}{(n^r)(r+3)} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e - 2}
Find the radius of convergence for the power series \( \sum_{n=1}^{\infty}\frac{ x^{6n+2}}{(1+\frac{1}{n})^{n^2}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{1/6}}
Let $A$ be an $R$-right module, $N$ be a submodule of an $R$-left module $M$, and let $\pi: M \rightarrow M/N$ be the natural epimorphism. Determine $\ker(\pi \otimes 1_A)$. Provide your answer in terms of $M$, $N$, and $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{N \otimes_R A}
Given that \(7^{-1} \equiv 55 \pmod{101}\), find \(49^{-1} \pmod{101}\), as a residue modulo 101. (Answer should be between 0 and 100, inclusive.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96}
In triangle ABC, angle C = 90 degrees and AC = BC = sqrt(2). Triangle ABC is rotated clockwise by 60 degrees around point A to the new position of triangle AB'C'. Connect C' to B. Calculate the length of C'B. Express your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3} - 1}
What is the smallest set of points required to intersect all lines in a projective geometry $PG(d,q)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{q^d - 1}{q - 1}}
Find a point on the curve \(r(t) = (1 - 2t)i + (t^2)j + \frac{t^3}{2}k\), where \(t > 0\), such that the tangent line at that point is parallel to the plane \(5x + y + z - 3 = 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-3, 4, 4)}
A specialty store sells three types of products: peach slices, rice crispy treats, and sesame snacks. The cost of each pack of peach slices is twice the cost of sesame snacks. The selling prices of each pack of peach slices, rice crispy treats, and sesame snacks are $20\%$, $30\%$, and $20\%$ higher than their respective costs. In May, the store sold peach slices, rice crispy treats, and sesame snacks in a ratio of $1:3:2$. The total profit from the three products was $25\%$ of the total cost. Find the ratio of the cost of each pack of rice crispy treats to the cost of each pack of sesame snacks.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4:3}
In a game of craps, if the first roll is neither a natural nor craps, the player can make an additional bet equal to their original one, betting that they will make their point before a seven appears. The payoff odds are as follows: 2:1 for a point of four or ten, 3:2 for a point of five or nine, and 6:5 for a point of six or eight. Calculate the expected winnings for the player if they make this additional bet when given the opportunity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Positive integers $a, b$, and $c$ have the property that $a^{b}, b^{c}$, and $c^{a}$ end in 4, 2, and 9, respectively. Compute the minimum possible value of $a+b+c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Given that the angle bisector of the lines represented by the equation \(a(x-1)^2+2h(x-1)(y-2)+b(y-2)^2=0\) is \(x+2y-5=0\), and the ratio \(\frac{b-a}{h}\) can be expressed as \(\frac{p}{q}\), where \(p\) and \(q\) are coprime integers, find the value of \(p+q\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
As shown in the diagram, there are 12 points on the circumference of a circle, dividing the circumference into 12 equal parts. How many rectangles can be formed using these equally divided points as the four vertices?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
K2) Let $n$ be a natural number. Roger has a square garden of size $(2 n+1) \times(2 n+1)$. He erects fences to divide it into rectangular beds. He wants exactly two horizontal $k \times 1$ beds and exactly two vertical $1 \times k$ beds for every even number $k$ between 1 and $2 n+1$, as well as a single square bed of size $1 \times 1$ when he is finished. How many different ways can Roger divide his garden?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n}}