inclusionAI/Ring-lite-rl-data huggingface dataset:
Consider a finite population from which $X_1, X_2, X_3, \ldots$ are sampled without replacement. The central limit theorem suggests that for i.i.d. random variables, the distribution of the sample mean approaches a normal distribution as the sample size $n$ increases. However, when sampling without replacement, independence does not hold, but the approximation to normality can still be valid when $n$ is small relative to the population size. Determine the value of $n$ for which the probability distribution of the sample mean is closest to the normal distribution described by the central limit theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
数列$\left\{ {{a}_{n}} \right\}$满足${{a1}}=1$，${{a}_{n+1}}=-a_{n}^{2}+c{{an}}-1$，若$\left\{ {{a}_{n}} \right\}$单调递减，则实数$c$的取值范围是__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\left( -\infty ,3 \right)$}
For a positive integer $ n$, let $ \theta(n)$ denote the number of integers $ 0 \leq x < 2010$ such that $ x^2 - n$ is divisible by $ 2010$. Determine the remainder when $ \displaystyle \sum_{n = 0}^{2009} n \cdot \theta(n)$ is divided by $ 2010$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
335
}
Let $\textbf{Z}_n$ be the set $\{0, 1, \ldots, n - 1\}$ with addition mod $n$. Consider subsets $S_n$ of $\textbf{Z}_n$ such that $(S_n + k) \cap S_n$ is nonempty for every $k$ in $\textbf{Z}_n$. Let $f(n)$ denote the minimal number of elements in such a subset. Determine the value of $$\lim_{n \to \infty} {\ln f(n) \over \ln n}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Polina wrote on the first page of her notebook \( n \) different positive
 integers. On the second page she wrote all pairwise sums of the
 numbers from the first page, and on the third - absolute values of
 pairwise differences of number from the second page. After that
 she kept doing same operations,i.e. on the page \( {2k} \) she wrote all
 pairwise sums of numbers from page \( {2k} - 1 \) , and on the page
 \( {2k} + 1 \) absolute values of differences of numbers from page \( {2k} \)
 At some moment Polina noticed that there exists a number \( M \)
 such that, no matter how long she does her operations, on every
 page there are always at most \( M \) distinct numbers.
 What is the biggest \( n \) for which it is possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
3
}
Determine the index of the subgroup $A \ast (ba)B(ba)^{-1}$ in the free product $A \ast B$, where $A$ and $B$ are non-trivial groups, and $a \in A$, $b \in B$ are non-identity elements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
\begin{quote}
Ted quite likes haikus,
poems with five-seven-five,
but Ted knows few words.

He knows $2n$ words
that contain $n$ syllables
for every int $n$.

Ted can only write
$N$ distinct haikus. Find $N$.
Take mod one hundred.
\end{quote}

Ted loves creating haikus (Japanese three-line poems with $5$, $7$, $5$ syllables each), but his vocabulary is rather limited. In particular, for integers $1 \le n \le 7$, he knows $2n$ words with $n$ syllables. Furthermore, words cannot cross between lines, but may be repeated. If Ted can make $N$ distinct haikus, compute the remainder when $N$ is divided by $100$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
28
}
Let \(x_1, x_2, \ldots, x_n\) be real numbers in arithmetic sequence, which satisfy \(|x_i| < 1\) for \(i = 1, 2, \dots, n,\) and \[|x_1| + |x_2| + \dots + |x_n| = 25 + |x_1 + x_2 + \dots + x_n|.\] What is the smallest possible value of \(n\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
Evaluate \[\frac 3{\log_5{3000^5}} + \frac 4{\log_7{3000^5}},\] giving your answer as a fraction in lowest terms.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{5}}
Determine the maximal size of a set of positive integers with the following properties:

$1.$ The integers consist of digits from the set $\{ 1,2,3,4,5,6\}$.

$2.$ No digit occurs more than once in the same integer.

$3.$ The digits in each integer are in increasing order.

$4.$ Any two integers have at least one digit in common (possibly at different positions).

$5.$ There is no digit which appears in all the integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Let\[X = \cos\frac{2\pi}7 + \cos\frac{4\pi}7 + \cos\frac{6\pi}7 + \cdots + \cos\frac{2006\pi}7 + \cos\frac{2008\pi}7.\]Compute $\Big|\lfloor 2008 X\rfloor\Big|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1004
}
What is the largest possible sum of the interior angles of a triangle on a torus? Consider the geometry of the torus and the placement of the triangle to determine this maximum sum.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{540^\circ}
Determine the number of elements in the set $D_n = \{ a^ib^j \mid \text{ order}(a)=n, \text{ order}(b)=2, a^ib = ba^{-i} \}$ that have order $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\phi(n)}
The number of people who purchased book A is twice the number of people who purchased book B. The number of people who purchased both books A and B is 500, which is twice the number of people who purchased only book B. Additionally, there is a third book, book C. The number of people who purchased only book C is equal to the number of people who purchased both books A and B. The total number of people who purchased any of these 3 books is 2,500. Given these constraints, what is the number of people x who purchased only book A?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1000}
## Task Condition

Find the derivative.

$y=\operatorname{arctg} \frac{\operatorname{tg} \frac{x}{2}+1}{2}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{\sinx+2\cosx+3}}
Calculate the limit of the following expression as $n$ approaches infinity, using the asymptotic expansion of the Barnes G-function:

$$\lim_{n\to\infty} \frac{G(n+\frac{11}{8})G(n+\frac{13}{8})^{2}G(n+\frac{15}{8})}{G(n+\frac{5}{2})^{2}G(n+\frac{7}{4})^{2}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
The product of two positive integers plus their sum is 71. The integers are relatively prime, and each is less than 15. What is the sum of the two integers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Divide the $n$ positive integers $1, 2, 3, \ldots, n$ ($n \in \mathbb{N}^*$) into two groups, such that no two numbers in each group sum up to a perfect square, and there are no identical numbers in the two groups. What is the maximum value of $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
A staircase has four steps. Calculate the total number of possible ways to climb the staircase, considering different step sequences as distinct. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
There exist positive integers $k$ and $3\nmid m$ for which
$$1 -\frac12 + \frac13 - \frac14 +...+ \frac{1}{53}-\frac{1}{54}+\frac{1}{55}=\frac{3^k \times m}{28\times 29\times ... \times  54\times 55}.$$Find the value $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Find all natural numbers \( n \) such that
$$
\min_{k \in \mathbf{N}}\left(k^{2}+\left\lfloor \frac{n}{k^{2}} \right\rfloor \right) = 1991 \quad (n \in \mathbf{N}).
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1024 \cdot 967 \leq n \leq 1024 \cdot 967 + 1023}
Two thousand points are given on a circle. Label one of the points 1. From this point, count 2 points in the clockwise direction and label this point 2. From the point labeled 2, count 3 points in the clockwise direction and label this point 3. Continue this process until the labels $1, 2, 3, \dots, 1993$ are all used. Some of the points on the circle will have more than one label and some points will not have a label. What is the smallest integer that labels the same point as 1993?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{118}
Let $b > 0$, and let $Q(x)$ be a polynomial with integer coefficients such that
\[Q(2) = Q(4) = Q(6) = Q(8) = b\]and
\[Q(1) = Q(3) = Q(5) = Q(7) = -b.\]
What is the smallest possible value of $b$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{315}
Three people are playing table tennis, where the player who loses a game gives up their place to the player who did not participate in that game. In the end, it turned out that the first player played 10 games, and the second player played 21 games. How many games did the third player play?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Compute the definite integral:

$$
\int_{0}^{\sqrt{2}} \frac{x^{4} \cdot d x}{\left(4-x^{2}\right)^{3 / 2}}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5 - \frac{3\pi}{2}}
Given a positive integer \( n \geq 3 \) and a sequence of \( n \) different real numbers \(\left(a_{1}, a_{2}, \cdots, a_{n}\right)\) whose sum is positive. If a permutation \(\left(b_{1}, b_{2}, \cdots, b_{n}\right)\) satisfies \( b_{1}+b_{2}+\cdots+b_{k} > 0 \) for every \( k = 1, 2, \cdots, n \), then this permutation is called good. Determine the minimum number of such good permutations.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(n-1)!}
A parliament consists of 30 members. Each pair of members are either friends or enemies, and each member has exactly 6 enemies. Every set of 3 members forms a 3-member committee. A committee is called a "good committee" if all three members are either mutual friends or mutual enemies. Determine the number of all good committees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1990}
Find the area enclosed by the parametric curve given by \( r(t) = (16\sin^3(t), 13\cos(t) - 5\cos(2t) - 2\cos(3t) - \cos(4t)) \) for \( 0 \le t \le 2\pi \). Use the integral \( \int_0^{2\pi} y \cdot x' \, dt \) and the hint: \( \int_0^{2\pi} \sin(nx)\sin(mx) \, dx = \int_0^{2\pi} \cos(nx)\cos(mx) \, dx = \begin{cases} \pi  & \text{for } m = n \\ 0 & \text{otherwise} \end{cases} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{180\pi}
Let  $ABC$  be a triangle with incenter  $I$ . Let  $M_b$  and  $M_a$  be the midpoints of  $AC$  and  $BC$ , respectively. Let  $B'$  be the point of intersection of lines  $M_bI$  and  $BC$ , and let  $A'$  be the point of intersection of lines  $M_aI$  and  $AC$ .

If triangles  $ABC$  and  $A'B'C$  have the same area, what are the possible values of  $\angle ACB$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
The cost price of an article is some percentage of the marked price. After allowing a discount of 25%, the gain percent is 17.1875%. What is the percentage of the cost price to the marked price?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64\%}
Vasya has a triangle with vertices labeled 1, 2, and 3. Rotating the triangle clockwise around vertex 1 by its angle 15 times returns it to its original position. Similarly, rotating around vertex 2 by its angle 6 times also returns it to its original position. Determine the minimum number of times, \( n \), that Vasya must rotate the triangle around vertex 3 by its angle to return it to its original position.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Evaluate the limit: \[ \lim_{n\rightarrow \infty}P\left(\sum_{i=1}^n X_i \leq \frac{n}{2}+ n^{3/4} \right) \] where \(X_1, X_2, \ldots, X_n\) are i.i.d. random variables with a uniform distribution on \([0, 1]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the points at which the function \( f(x,y) = \begin{cases} xy\sin\left(\frac{1}{x^2+y^2}\right), & (x,y) \neq (0,0) \\ 0, & (x,y) = (0,0) \end{cases} \) is differentiable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}^2}
The price of an article has undergone three successive discounts: first by 15%, then by 20%, and finally by 25%. After these three discounts are applied sequentially, the final cost of the article is Rs. 250. What was the original cost of the article?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{490.20}
Find the relationship between the initial conditions \(x(0) = x_0\) and \(\dot{x}(0) = y_0\) that ensures the solution \(x(t) = Ae^{k_1t} + Be^{-k_2t}\) of the differential equation approaches zero as \(t \to \infty\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y_0 = -k_2 x_0}
In acute triangle $\Delta ABC$, $O$ is the circumcenter, $\angle A=45^\circ$. If $\frac{\cos B}{\sin C}\overset{\rightarrow}{AB}+\frac{\cos C}{\sin B}\overset{\rightarrow}{AC}=2m\overset{\rightarrow}{AO}$, find the value of $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{2}}{2}}
Let \(Q\) be a set of permutations of the numbers \(1, 2, \ldots, 100\) such that for any two numbers \(a\) and \(b\) with \(1 \leq a, b \leq 100\), the number \(a\) appears to the left of \(b\) and adjacent to \(b\) in at most one permutation in \(Q\). Determine the largest possible number of permutations in the set \(Q\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Several schools took part in a tournament. Each player played one match against each player from a different school and did not play anyone from the same school. The total number of boys taking part differed from the total number of girls by 1. The total number of matches with both players of the same sex differed by at most one from the total number of matches with players of opposite sex. What is the largest number of schools that could have sent an odd number of players to the tournament?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Evaluate the integral \( \int_0^\pi e^{ix} \sqrt{1 - e^{2ix}} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{i\pi}{2}}
A train crosses a platform in 40 seconds while it crosses a signal pole in 20 seconds. The length of the platform is the same as the length of the train. What is the length of the train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
You have a Lego block of length 10 and three additional Lego blocks of lengths 3, 4, and 5. These shorter blocks are randomly placed on the block of length 10 such that their boundaries are within the boundaries of the block of length 10. The position of a short block is defined by the left-most knob it covers on the 10-block. The block of length 3 can be placed in positions 1 to 8, the block of length 4 in positions 1 to 7, and the block of length 5 in positions 1 to 6, all with equal probability. The blocks are placed independently. Let $X$ be the number of knobs on the large block that are covered by all three shorter blocks. Calculate the expected value $E[X]$. \(X \in \{0, 1, 2, 3\}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{6}{7}}
Find the optimal solution to the optimization problem:
\[\begin{aligned}
&\min_{\mathbf{x} \in \mathbb{R}^n} && \|\mathbf{B}\mathbf{x}\|_1 \\
&\ \ \text{s.t.} && \|\mathbf{x}\|_2 \leq 1
\end{aligned}\]
where \(\mathbf{B} \in \mathbb{R}^{n \times n}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbf{0}}
25. Let $\dot{P}$ be a 40 -sided convex polygon. Find the number of triangles $S$ formed by the vertices of $P$ such that any two vertices of $S$ are separated by at least two other vertices of $P$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7040}
1. Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$. What is the smaller part of the volume separated from it by a plane passing through the midpoint of edge $B C$, the center of face $C D D_{1} C_{1}$, and point $X$ on edge $A A_{1}$ such that $A X: A_{1} X=1: 3 ?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{25}{96}}
Barbara has 9 candies. She buys 18 more and shares them with her 6 friends. Each friend receives an equal amount of candies. If each friend also receives some of Barbara's original candies, how many candies does each friend get?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find a vector potential $G$ such that $\nabla \times G = (x, -y, 0)$. Provide your answer as a vector $G$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 0, xy)}
From a point P on the surface of a sphere with radius R, draw three mutually perpendicular chords PA, PB, and PC. Find the value of $PA^2 + PB^2 + PC^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4R^2}
Consider the stochastic differential equation \(dZ_t = aZ_t dt + \sigma Z_t dB_t\), where \(B_t\) is a Brownian motion, \(a = 0.03\), and \(\sigma = 0.3\). Given that \(Z_0 = 100\), find the probability \(P(Z_1 > 103.5)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.4345}
Jason is counting the number of cars that pass by his window. He notices that there are twice as many blue cars as there are red cars. Furthermore, there are three times as many red cars as there are orange cars. He also notices that there are half as many yellow cars as there are orange cars, but three times as many yellow cars as there are purple cars. Lastly, there are five times as many green cars as there are purple cars. If Jason counted a total of 987 cars, how many of them were purple?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Equilateral $\\triangle ABC$ has side length $\\sqrt{111}$. There are four distinct triangles $AD_1E_1$, $AD_1E_2$, $AD_2E_3$, and $AD_2E_4$, each congruent to $\\triangle ABC$, with $BD_1 = BD_2=\\sqrt{11}$. Find $\\sum^4_{k=1}(CE_k)^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{677}
Compute
$$\frac{lcm (1,2, 3,..., 200)}{lcm (102, 103, 104, ..., 200)}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{101}
Given that the focus of the parabola $y^{2}=8x$ is $F$, two perpendicular lines $l_{1}$ and $l_{2}$ are drawn passing through point $F$. Let these lines intersect the parabola at points $A$, $B$ and $D$, $E$, respectively. The minimum area of quadrilateral $ADBE$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{128}
Compute the limit: \[ \lim_{x \to \infty} \int_{0}^{\pi} \frac{\sin(t)}{1+\cos^{2}(xt)} \, \mathrm{d}t. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Two distinct similar rhombi share a diagonal. The smaller rhombus has an area of $1$, and the larger rhombus has an area of $9$. Compute the side length of the larger rhombus.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Define $$p(x)= \sum_{i=1}^{n+1}2^{i}\cdot\frac{\prod_{j=1,j\neq i}^{n+1}(x-j)}{\prod_{j=1,j\neq i}^{n+1}(i-j)}.$$ Find $p(n+2)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n+2}-2}
Lucy was born on Wednesday, December 1st, 2004. That Wednesday was the first day of her life. Her parents held a party for her on the $1000$th day of her life. On which day of the week was the party?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\text{Monday}}
Solve the equation \(x = e^{-1/x}\) for \(x\) analytically.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{W(-1)}}
Compute the cross-product term \(d(x_1)d(x_2)\) for the following independent Wiener processes:
\[ dx_1 = a_1 dt + b_1 dw_1(t) \]
\[ dx_2 = a_2 dt + b_2 dw_2(t) \]
where \(w_1\) and \(w_2\) are independent Wiener processes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine whether there exists a subfield of the real numbers that is both uncountable and locally compact.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}}
Given a cube, on each edge, write a number, either $1$ or $-1$. For each face of the cube, multiply the four numbers on the edges of this face and write the product on this face. Finally, add all the eighteen numbers that were written on the edges and faces of the cube.

What is the smallest possible sum that can be obtained?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-12}
For non-negative real numbers $x_1, x_2, \ldots, x_n$ which satisfy $x_1 + x_2 + \cdots + x_n = 1$, find the largest possible value of $\sum_{j = 1}^{n} (x_j^{4} - x_j^{5})$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{12}}
What relationship must hold between the edges of a cuboid for it to be divided into two congruent cuboids that are similar to the original one?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 : \sqrt[3]{2} : \sqrt[3]{4}}
In a village, there are three age groups of workers: young, middle-aged, and old. The work capacities of young and old workers are only 75% of middle-aged workers.

Eight middle-aged men can dig a pit in 20 days. If a middle-aged man works half as much again as a boy, determine the number of days it would take for the following team to dig a similar pit: 2 young men, 2 middle-aged men, 2 old men, 5 young boys, 5 middle-aged boys, and 5 old boys.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Determine the dimension of the tangent space at a point where multiple circles intersect in \(\mathbb{R}^2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the extremals for the functional \( J(y) = \int_{0}^{1}\left[\dot{y}^{2}\right]\mathrm{d}t \) subject to the constraints \( y(0) = 1 \), \( y(1) = 4 \), and \( \int_{0}^{1}y\mathrm{d}t = 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3t^2 + 1}
Given an element \( \alpha \) in a field with characteristic 2, and its minimal polynomial \( f(x) = x^9 + x + 1 \) with respect to \( GF(2) \), determine the order of \( \alpha \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{73}
考察集合$A = \{ 1 , 2 , 3 , 4 , \cdots ,2 ^ { n } \}$，其中，$n \geq 2 .$如果集合$A$中两个元素的和是2的幂，那么，它们之中的一个恰好属于子集$B$.求集合$A$中子集$B$的个数.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2 ^ { n + 1 }$}
How many ways are there to partition the set $\left\{2^{0}, 2^{1}, 2^{2}, \cdots, 2^{2005}\right\}$ into two non-empty disjoint subsets $A$ and $B$, such that the equation $x^{2} - S(A)x + S(B) = 0$ has integer roots, where $S(M)$ denotes the sum of all elements in the set $M$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1003}