inclusionAI/Ring-lite-rl-data huggingface dataset:
Evaluate the integral \( \int_{C_r} \frac{100z^{100}+99z^{99}}{z^{100}+z^{99}+1} \, dz \) using the Cauchy integral theorem, where \(C_r\) is a contour that encloses all the roots of the denominator.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2\pi i}
Two cars start from the opposite places of a main road, 150 km apart. The first car runs for some distance and takes a right turn and then runs 15 km. It then turns left and then runs for the same distance as before and then takes the direction back to reach the main road. In the meantime, due to a minor breakdown, the other car has run only 62 km along the main road. The distance between the two cars at this point is 38 km. How far did the first car run before taking the right turn?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Let $X_i \sim N(0,1)$ for $0 \leq i \leq n$ be independent standard normal distributions. Define $Y_n := \frac{\sqrt{n} X_0}{\sqrt{(\sum_{i=1}^n X_i^2)}}$. Determine the limiting distribution function for $Y_n$ as $n$ approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Phi(x)}
Determine the asymptotic lower bound of the sum \( S = \frac{3}{2} + \sum_{k=3}^{n} \frac{n!}{k(n-k)!n^k} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Omega(\log n)}
Compute the rank of the $\mathbb{Q}$ vector space $\text{Hom}_R(M,R/(x^2))$, where $R=\mathbb{Q}[x]$ and $M$ is the cokernel of the map $R^2 \rightarrow R^2$ given by the matrix \(\begin{bmatrix} x-1 & 0 \\ 1-x & x^2 \end{bmatrix}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let \( S = 2^3 + 3^4 + 5^4 + 7^4 + \cdots + 17497^4 \) be the sum of the fourth powers of the first 2014 prime numbers. Find the remainder when \( S \) is divided by 240.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{93}
The positive divisors of a positive integer \( n \) are ordered as \( 1 = d_1 < d_2 < \cdots < d_k = n \). Suppose \( d_7^2 + d_{15}^2 = d_{16}^2 \). Find all possible values of \( d_{17} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
Find the derivative of \(\tan^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)\) with respect to \(\tan^{-1}\left(\frac{2x\sqrt{1-x^2}}{1-2x^2}\right)\) at \(x=0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
The width of rectangle $ABCD$ is twice its height, and the height of rectangle $EFCG$ is twice its width. The point $E$ lies on the diagonal $BD$. Which fraction of the area of the big rectangle is that of the small one?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$\frac{4}{25}$}
Does there exist a function that has an infinite number of non-zero derivatives, starts at negative infinity, and approaches zero as \( x \) goes to infinity?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-e^{-x}}
Let $G$ be a group where $a$ and $b$ are non-identity elements such that $ab = b^2a$. If the subgroup of $G$ generated by $a$ has order 3, determine the order of the subgroup of $G$ generated by $b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Find three positive integers $a, b, c$ whose sum is $407$, and whose product (when written in base $10$) ends in six $0$'s.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
a = 125, b = 250, c = 32}
Consider the set of integers \( \mathbb{Z} \) with the usual less-than relation. Determine whether the well-founded part of this set, \( W(\prec) \), is the empty set.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
Determine the minimum capacity required for each of 10 boxes such that in 90% of the games, no box overflows. In each of the 50 trials, 4 boxes are selected at random, and one ball is placed into each selected box. There are 200 balls in total. What is the minimum capacity of each box?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
A spherical shell of mass $M$ and radius $R$ is completely filled with a frictionless fluid, also of mass M. It is released from rest, and then it rolls without slipping down an incline that makes an angle $\theta$ with the horizontal. What will be the acceleration of the shell down the incline just after it is released? Assume the acceleration of free fall is $g$. The moment of inertia of a thin shell of radius $r$ and mass $m$ about the center of mass is $I = \frac{2}{3}mr^2$; the momentof inertia of a solid sphere of radius r and mass m about the center of mass is $I = \frac{2}{5}mr^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$\frac{3}{4} g \sin \theta$

}
Suppose $g(x)$ is a differentiable, real-valued, periodic function with period $a$ such that for all $u$, $\int_u^{u+a} g(x)\,dx=0$. Determine whether it is true that $\int_u^{u+a}g(x+g(x))\,dx=0$ for all $u$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ and two lines $l_1: y = \frac{1}{2}x$ and $l_2: y = -\frac{1}{2}x$, parallel lines to $l_1$ and $l_2$ pass through a point $P$ on the ellipse and meet $l_1$ and $l_2$ at points $M$ and $N$, respectively. If $|MN|$ is a constant value, then the value of $\sqrt{\frac{a}{b}}$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
# Problem 7. Option 1.

Given a triangle with sides 6, 8, and 10. Find the length of the shortest segment connecting points on the sides of the triangle and dividing it into two equal areas.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given are the sequences
\[ (..., a_{-2}, a_{-1}, a_0, a_1, a_2, ...);  (..., b_{-2}, b_{-1}, b_0, b_1, b_2, ...); (..., c_{-2}, c_{-1}, c_0, c_1, c_2, ...)\]
of positive real numbers. For each integer $n$ the following inequalities hold:
\[a_n \geq \frac{1}{2} (b_{n+1} + c_{n-1})\]
\[b_n \geq \frac{1}{2} (c_{n+1} + a_{n-1})\]
\[c_n \geq \frac{1}{2} (a_{n+1} + b_{n-1})\]
Determine $a_{2005}$, $b_{2005}$, $c_{2005}$, if $a_0 = 26, b_0 = 6, c_0 = 2004$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2004, 26, 6)}
For a real-valued function \( f\left( {x,y}\right) \) of two positive real variables \( x \) and \( y \) ,define \( f \) to be linearly bounded if and only if there
 exists a positive number \( K \) such that \( \left| {f\left( {x,y}\right) }\right|  < K\left( {x + y}\right) \) for all positive \( x \) and \( y \) . Find necessary and sufficient conditic
 on the real numbers \( \alpha \) and \( \beta \) such that \( {x}^{\alpha }{y}^{\beta } \) is linearly bounded.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha + \beta=1, \alpha,\beta \geq 0}
Bernado and Silvia play the following game. An integer between 0 and 999, inclusive, is selected and given to Bernado. Whenever Bernado receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds 50 to it and passes the result to Bernado. The winner is the last person who produces a number less than 1000. Let $N$ be the smallest initial number that results in a win for Bernado. What is the sum of the digits of $N$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
7
}
Let's define a point on the plane $(u, v)$ as lying between the parabolas \( y = f(x) \) and \( y = g(x) \) if \( f(u) \leq v \leq g(u) \). Find the smallest real number \( p \) for which the following statement is true: any line segment whose endpoints and midpoint lie between the parabolas \( y = x^2 \) and \( y = x^2 + 1 \), lies entirely between the parabolas \( y = x^2 \) and \( y = x^2 + p \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9}{8}}
In a card game with 101 cards, each card has a unique positive integer from 1 to 101 written on it. You win if the last card you turn over is the highest in the deck. What is the optimal number of cards to turn over from the first half of the deck to maximize your probability of winning?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
Find the limit of the sequence defined by the recurrence relation \(x_{n+1} = x_n(2 - ax_n)\), where \(a\) is a positive real number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{a}}
13. (10 points) A class of 30 students goes to the movies, and their student IDs are $1, 2, \cdots, 30$; the movie tickets they hold are exactly the 1st, 2nd, ..., 30th seats in a row. Now, the movie tickets are to be distributed to these students according to the following requirements: for any two students, A and B, if A's student ID can be divided by B's student ID, then A's movie ticket number can also be divided by B's movie ticket number. How many different ways are there to distribute the movie tickets? $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Given an isosceles triangle \( ABC \) with \( AB = BC \). Points \( K \), \( L \), and \( M \) are marked on the sides \( AB \), \( BC \), and \( AC \) respectively such that \(\angle AKM = 90^\circ\), \(\angle BLK = 90^\circ\), and \( KM = KL \). What is the measure of angle \( CML \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
Let $f : \mathbb{N} \to \mathbb{N}$ be a function such that the following conditions hold:

$\qquad\ (1) \; f(1) = 1.$

$\qquad\ (2) \; \dfrac{(x + y)}{2} < f(x + y) \le f(x) + f(y) \; \forall \; x, y \in \mathbb{N}.$

$\qquad\ (3) \; f(4n + 1) < 2f(2n + 1) \; \forall \; n \ge 0.$

$\qquad\ (4) \; f(4n + 3) \le 2f(2n + 1) \; \forall \; n \ge 0.$


Find the sum of all possible values of $f(2023)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1012}
Consider a surface in \(\mathbb{R}^3\) with the first fundamental form given by \(ds^2=E(u)du^2+G(v)dv^2\). Determine whether the Gaussian curvature of this surface vanishes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the improper integral \(I=\int_{-\infty}^\infty\frac{x\sin(\pi x)}{x^2+2x+5}dx\) using complex analysis techniques.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\pi e^{-2\pi}}
One day Jason finishes his math homework early, and decides to take a jog through his neighborhood. While jogging, Jason trips over a leprechaun. After dusting himself off and apologizing to the odd little magical creature, Jason, thinking there is nothing unusual about the situation, starts jogging again. Immediately the leprechaun calls out, "hey, stupid, this is your only chance to win gold from a leprechaun!"

Jason, while not particularly greedy, recognizes the value of gold. Thinking about his limited college savings, Jason approaches the leprechaun and asks about the opportunity. The leprechaun hands Jason a fair coin and tells him to flip it as many times as it takes to flip a head. For each tail Jason flips, the leprechaun promises one gold coin.

If Jason flips a head right away, he wins nothing. If he first flips a tail, then a head, he wins one gold coin. If he's lucky and flips ten tails before the first head, he wins $\textit{ten gold coins.}$ What is the expected number of gold coins Jason wins at this game?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
1
}
Problem 4. Find the number of all natural numbers $n, 4 \leq n \leq$ 1023, such that their binary representations do not contain three consecutive equal digits.

Emil Kolev

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{228}
Observing the temperatures recorded in Cesenatico during the December and January, Stefano noticed an interesting coincidence: in each day of this period, the low temperature is equal to the sum of the low temperatures the preceeding day and the succeeding day.
Given that the low temperatures in December $3$ and January $31$ were $5^\circ \text C$ and $2^\circ \text C$ respectively, find the low temperature in December $25$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3}
Reduce the lambda calculus expression $$(\lambda x. (\lambda y. x \ y) \ x) \ (\lambda z.p)$$ to its simplest form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p}
Determine the value of $a$ such that the expression $n^4 + a$ can be factored into two expressions with integer coefficients for every natural number $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
On the island of Misfortune, there live knights who always tell the truth and liars who always lie. One day, 2023 natives, among whom $N$ are liars, stood in a circle and each said, "Both of my neighbors are liars". How many different values can $N$ take?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{337}
Determine the range of the function \( \frac{f(x)}{x} \) as \( x \to \infty \), where \( f(x) = 2x - \sigma(x) \) and \( \sigma(x) \) is the sum-of-divisors function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 1]}
Compute the homology group $H_2(\mathbb{S}^3/G; \mathbb{Z})$ for a finite subgroup $G$ of $SO(4)$ acting freely on $\mathbb{S}^3$. Is this homology group zero?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Consider the set of solutions of the equation
 \( {x}^{2} + {y}^{3} = {z}^{2}. \)
 in positive integers. Is it finite or infinite?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
Infinite

}
Determine the range of values for \( k \) such that the inequality \[ \int_0^x \frac {dt}{\sqrt {(3 + t^2)^3}} \geq k\int _0^x \frac {dt}{\sqrt {3 + t^2}} \] holds for all \( 0 \leq x \leq 1 \). You may use \( \ln 3 = 1.10 \) if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3 \ln 3}}
若存在正整数 $a, b, c$ 满足 $a+b+c=407,10^{n} \mid a b c$ ，求 $n$ 的最大值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let \( G = \mathbb{Z}_{p^2} \times \mathbb{Z}_{p^2} \), where \( p \) is a prime number. Determine the number of distinct subgroups of order \( p^2 \) in \( G \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p^2 + p + 1}
Determine the interval of convergence for the integral \( \int_1^{+\infty} \frac{x^{\alpha} \cdot \sin x}{1+x^3} \, dx \). Consider how the parameter \( \alpha \) affects the convergence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 3)}
How many positive integers $n \leq 100$ satisfy $\lfloor n\pi \rfloor = \lfloor (n-1)\pi \rfloor + 3$? Here $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$; for example, $\lfloor \pi \rfloor = 3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{86}
Let \( x, y, z, w \) be four consecutive vertices of a regular \( A \)-gon. If the length of the line segment \( xy \) is 2 and the area of the quadrilateral \( xyzw \) is \( a + \sqrt{b} \), find the value of \( B = 2^a \cdot 3^b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
Four numbers, 2613, 2243, 1503, and 985, when divided by the same positive integer, yield the same remainder (which is not zero). Find the divisor and the remainder.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
Let $d(n)$ denote the number of positive integers that divide $n$, including $1$ and $n$. For example, $d(1)=1,d(2)=2,$ and $d(12)=6$. (This function is known as the divisor function.) Let\[f(n)=\frac{d(n)}{\sqrt [3]n}.\]There is a unique positive integer $N$ such that $f(N)>f(n)$ for all positive integers $n\ne N$. What is the sum of the digits of $N?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Before Harry Potter died, he decided to bury his wand in one of eight possible locations (uniformly at random). A squad of Death Eaters decided to go hunting for the wand. They know the eight locations but have poor vision, so even if they're at the correct location they only have a  $50\%$  chance of seeing the wand. They also get tired easily, so they can only check three different locations a day. At least they have one thing going for them: they're clever. Assuming they strategize optimally, what is the expected number of days it will take for them to find the wand?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{253}{56}}
Using 6 different colors to paint the 4 cells in the picture, with each cell painted in one color, and requiring that at most 3 colors are used and no two adjacent cells have the same color, how many different coloring methods are there? (Answer using a number.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{390}
For which $n$-gons can a regular $n$-gon be placed on a lined sheet of paper such that all vertices lie on the lines?

(The lines are parallel and evenly spaced.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3, 4, 6}
Find the residue of the function \( f(z) = z^{n-1} e^{\frac{pz}{2}} e^{\frac{p}{2z}} \) at the essential singularity \( z = 0 \), where \( n \in \mathbb{N} \) and \( p \in \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{I_n(p)}
A pedestrian and a cyclist are moving in one direction on a highway, while a cart and a car are moving in the opposite direction. All participants are moving at constant speeds (each with their own speed). The cyclist first overtook the pedestrian, then after some time met the cart, and then after the same amount of time met the car. The car first met the cyclist, then after some time met the pedestrian, and then after the same amount of time overtook the cart. The cyclist overtook the pedestrian at 10 o'clock, and the pedestrian met the car at 11 o'clock. When did the pedestrian meet the cart?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10:40}
A police car is moving down a narrow street at 90 km/h, with a blue light on top rotating counterclockwise at one rotation per second. The buildings are 3 meters from the car on the right. Calculate the speed of the light beam on the wall when it is 6 meters ahead of its source.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30\pi - 25}
Determine the number of unique Tsuro tiles that can exist. Each tile has 8 entry points, and each point is connected to exactly one other point. Consider the uniqueness of tiles with respect to rotations.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
Calculate the integral \( I = \iint_R |xy|\,dx\,dy \) over the region \( R \) enclosed by the curves \( x^2+4y^2 = 1 \) and \( x^2+y^2 = 1 \) using polar coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{8}}
Find the limit \( \lim_{{n \to \infty}} x_n \) for the convergent sequence \( \{x_n\} \) defined by the equation \( x_n^3 - (n^2 + 3n)x_n - 3n^2 = 0 \) for all \( n \in \mathbb{N} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3}
In a $4\times 4$ grid, there are $8$ blue tiles and $8$ white tiles. Determine the number of distinct ways to arrange the tiles in the grid, considering rotations and reflections as equivalent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1674}
Find the base with the smallest cardinality for the topological space (X, \(\tau\)), where X = \(\mathbb{R}^2\) and \(\tau = \{\emptyset, \mathbb{R}^2, \{(x, y) \in \mathbb{R}^2 \mid x^2+y^2<r \}, r \in \mathbb{R}, r>0\}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\aleph_0}
For a given positive integer $k$, consider a game played on an $n \times n$ board where $n \geq k$. The pigeon and the seagull take turns, with the pigeon going first. The pigeon chooses $m$ grids to lay an egg in each, and the seagull chooses a $k \times k$ subgrid to eat all the eggs inside it. The pigeon wins if every grid on the board eventually contains an egg; otherwise, the seagull wins. Determine all values of $m$ for which the pigeon can guarantee a win.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m \geq k^2 + 1}
Consider a positive integer, $N = 9 + 99 + 999 + ... +\underbrace{999...9}_{2018}$.
How many times does the digit $1$ occur in its decimal representation?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2016
}
Let \( X \sim N(0,1) \). Determine the distribution of \( \Phi(X) = \int_{-\infty}^X {\exp\left(-{z^2}{/2}\right) \over \sqrt{2\pi}} \, \mathrm{d}z \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{U(0,1)}
Find the sum of the series \(\sum_{n=0}^\infty (z^{2^{n+1}} - z^{2^n})\) for \(|z| < 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-z}
Evaluate the limit of the sequence of integrals: \[ \lim _{n \rightarrow \infty} \int_{-\infty}^\infty \frac{dx}{n(e^{x^2}-1) +1/n} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
What is the correct order of the fractions $\frac{15}{11}, \frac{19}{15},$ and $\frac{17}{13},$ from least to greatest?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{19}{15}<\frac{17}{13}<\frac{15}{11}}
Consider an urn with $k$ balls, numbered from $1$ to $k$. A step involves drawing one ball uniformly at random, removing all balls with labels greater than the drawn ball's number, and returning the drawn ball to the urn. The process continues until the ball labeled $1$ is drawn. What is the expected number of steps needed to draw the ball with label $1$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + \sum_{i=1}^{k-1} \frac{1}{i}}
Find the rational term for $b=11$ in the series \( \sum_{n=0}^\infty \frac{1}{n^2+bn+c} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{5}}
Calvin has a bag containing 50 red balls, 50 blue balls, and 30 yellow balls. Given that after pulling out 65 balls at random (without replacement), he has pulled out 5 more red balls than blue balls, what is the probability that the next ball he pulls out is red?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\( \frac{9}{26} \)}
Determine the smallest order greater than 1 for which two orthogonal complete Latin squares exist. A Latin square of order $n$ is an $n \times n$ array filled with $n$ distinct symbols $\{s_1, \ldots, s_n\}$, such that each symbol appears exactly once in each row and each column. Two Latin squares, denoted $L_{ij}$ and $G_{ij}$, are orthogonal if each of the $n^2$ pairs $(L_{ij}, G_{ij})$ is distinct. A Latin square is complete if each of the $n(n-1)$ pairs $(L_{ij}, L_{i+1,j})$ and $(L_{ij}, L_{i,j+1})$ is distinct.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}