inclusionAI/Ring-lite-rl-data huggingface dataset:
Find the largest prime number less than $2008$ that is a divisor of some integer in the infinite sequence \[ \left\lfloor\frac{2008^1}{1}\right\rfloor, \left\lfloor\frac{2008^2}{2}\right\rfloor, \left\lfloor\frac{2008^3}{3}\right\rfloor, \left\lfloor\frac{2008^4}{4}\right\rfloor, \ldots. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2003}
In a 7x7 grid, each unit square can be part of a Swiss cross shape consisting of 5 unit squares. Determine the minimum number of squares that must be marked on the grid so that every possible placement of a Swiss cross covers at least one marked square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
John watches a TV show and they announce they are going to do 1 more season. Each season has a certain number of episodes except for the last season which is 4 episodes longer. There were 9 seasons before the announcement. If each episode is .5 hours, it will take 112 hours to watch them all after the last season finishes. How many episodes are there in each season?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
In convex quadrilateral \(EFGH\), \(\angle E = \angle G\), \(EF = GH = 150\), and \(EH \neq FG\). The perimeter of \(EFGH\) is 580. Find \(\cos E\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{14}{15}}
Calculate the work done by the force field $F(x, y) = (-y^2, x^2)$ along the boundary of the rectangle defined by the lines $x - y = 0$, $x - y = 1$, $x + y = 1$, and $x + y = 2$, where the path is taken in the positive direction (counter-clockwise) using Green's theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3/2}
甲、乙两人进行乒乓球比赛，约定每局胜者得 1 分，负者得 0 分，比赛进行到有一人比对方多 2 分或打满 6 局时停止。设甲在每局中获胜的概率为 \(\frac{2}{3}\)，乙在每局中获胜的概率为 \(\frac{1}{3}\)，且各局胜负相互独立。求比赛停止时已打局数 \(\xi\) 的期望 \(E \xi\)。答案形式为\(\frac{k}{m}\)，请给出k+m的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{347}
A sequence of positive integers with $n$ terms satisfies $\sum_{i=1}^{n} a_i = 2007$. Find the least positive integer $n$ such that there exist some consecutive terms in the sequence with their sum equal to $30$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1018}
Let $G=S_5$. Consider the set $L$ of subgroups of $G$ that are isomorphic to the non-cyclic group of order 4. Let us call two subgroups $H$ and $K$ belonging to $L$ as equivalent if they are conjugate (that is $gHg^{-1}=K$ for some $g\in G$). How many equivalence classes are there in $L$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
a side of beef lost 35 percent of its weight in processing . if the side of beef weighed 560 pounds after processing , how many pounds did it weigh before processing ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{861.54}
Two boys start moving from the same point A on a circular track but in opposite directions. Their speeds are 5 ft. per second and 9 ft. per second. If they start at the same time and finish when they first meet at the point A again, calculate the number of times they meet, excluding the start and finish. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Given the function f(x) = 7 + a^(x-1), where a > 0 and a!= 1, find the coordinates of the fixed point P that the graph of this function passes through for any valid value of a. Express your answer as an ordered pair (x, y).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, 8)}
19. Fill in the following squares with $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ respectively, so that the sum of the two five-digit numbers is 99999. How many different addition equations are there? (Consider $a+b$ and $b+a$ as the same equation)
$\square \square \square \square \square+$ $\square$ $=99999$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1536}
Determine the maximum power of 2 that completely divides \(3^{1024} - 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Given two circles ${C}_{1}:(x-a)^{2}+{y}^{2}=36$ and ${C}_{2}:{x}^{2}+(y-2)^{2}=4$ are tangent internally, then $a=\_\_\_\_\_\_.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±2\sqrt{3}}
Let \( f(x) = x^3 - x^2 - 3x - 1 \) and \( h(x) = \frac{f(x)}{g(x)} \) be a rational function such that:

(a) \( h(x) \) is continuous everywhere except at \( x = -1 \).
(b) \( \lim_{x \to \infty} h(x) = \infty \).
(c) \( \lim_{x \to -1} h(x) = \frac{1}{2} \).

Find \( \lim_{x \to 0} (3h(x) + f(x) - 2g(x)) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{39}{4}}
When triangle $EFG$ is rotated by an angle $\arccos(_{1/3})$ around point $O$, which lies on side $EG$, vertex $F$ moves to vertex $E$, and vertex $G$ moves to point $H$, which lies on side $FG$. Find the ratio in which point $O$ divides side $EG$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3:1}
Evaluate the limit: $$\lim_{x\rightarrow\infty} \frac{1}{x}\left(\frac{\Gamma(2x-1)}{\Gamma(x-1)}\right)^{1/x}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{e}}
Evaluate the limit: \[ \lim_{k \rightarrow \infty} \int_{-\infty}^\infty \frac{1}{1+x^2+\cdots+x^{2k}} \, \text{dx} = \lim_{k \rightarrow \infty} \int_{-\infty}^\infty \frac{1}{\sum_{n=0}^k x^{2n}} \, \text{dx}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{3}}
Point $P(2,16)$ and $Q(\log_{2}3,t)$ both lie on the graph of the same exponential function. Find $t$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
In an election, there are three candidates - A, B, and C. Candidate A gets 50% of the votes, candidate B gets 35% of the votes, and candidate C gets 15% of the votes. Candidate A is elected by a majority of 1400 votes over candidate B. What is the total number of votes polled?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9333}
Bobby needs to buy a new pair of fancy shoes. He decides to go to a cobbler and get them handmade. The cobbler charges $250 to make the mold. He then charges $75 an hour for 8 hours to make the shoes. The cobbler agrees to charge a certain percentage of the cost for work to make the shoes, since it is his first pair of shoes. Bobby paid $730. What percentage of the cost for work did the cobbler charge?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20\%}
A point is randomly selected on a plane, where its Cartesian coordinates are integers with absolute values less than or equal to 4, and all such points are equally likely to be chosen. Calculate the probability that the selected point is at most 2 units away from the origin. Express your answer as a fraction in simplest form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{13}{81}}
Given the hyperbola $C$: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$, $F_{1}$, $F_{2}$ are its left and right foci, point $P$ is a point on the right branch of the hyperbola, point $M$ is the center of the circle, circle $M$ is the inscribed circle of triangle $PF_{1}F_{2}$, $PM$ line intersects the $x$-axis at the point $(1, 0)$, and is parallel to one of the asymptotes of the hyperbola and is at a distance of $\frac{\sqrt{2}}{2}$. Calculate the eccentricity of the hyperbola $C$. Express your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Find the closure of the set of irrational numbers in the cocountable topology on the real numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}}
How many nine-digit numbers composed of eight 3's and one 0 satisfy the condition that they leave a remainder of 1 when divided by 4? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
An geometric sequence $\{a_n\}$ has 20 terms, where the product of the first four terms is $\frac{1}{128}$, and the product of the last four terms is 512. The product of all terms in this geometric sequence is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Given the set A = {1, 2, 3, 4, 5, 6}, count the number of bijections f: A -> A that satisfy the condition f(f(f(x))) = x for all x in A. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{81}
Samson writes down the number 123456789 on a piece of paper. He can insert multiplication signs between any two adjacent digits, any number of times at different places, or none at all. By reading the digits between the multiplication signs as individual numbers, he creates an expression made up of the products of these numbers. For example, 1234$\cdot$56$\cdot$789. What is the maximum possible value of the resulting number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{123456789}
Given the equation \(1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{23} = \frac{a}{23!}\), find the value of \(a \mod 13\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Calculate the probability \( P(p=0.5|X=7) \) given that \( X \sim \text{Bin}(8, p) \) and \( p \) is either 0.5 or uniformly distributed over [0.5, 1]. Use Bayes' Rule to find the solution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{36}{287}}
Maria buys computer disks at a price of 5 for $6 and sells them at a price of 4 for $7. Find how many computer disks Maria must sell to make a profit of $120.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{219}
Suppose $X, Y$ are independent and identically distributed random variables with $X,Y\sim N(0,1)$. What is $P(X > 0 \mid X + Y > 0)?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3/4}
There are 10 consecutive natural numbers written on the board. What is the maximum number of them that can have a digit sum equal to a perfect square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
We are given a coin of diameter $\frac{1}{2}$ and a checkerboard of $1 \times 1$ squares with an area of $2010 \times 2010$. We toss the coin such that it lands completely on the checkerboard. If the probability that the coin doesn't touch any of the lattice lines is $\frac{a^2}{b^2}$, where $\frac{a}{b}$ is a reduced fraction, find $a + b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6029}
Compute the genus of the curve given by the intersection \( \{X^4+Y^4+Z^4=0\} \cap \{X^3+Y^3+(Z-tW)^3=0\} \subset \mathbb{P}^3 \). Use the adjunction formula and the fact that \( K_{\mathbb{P}^3}=O(-4) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
A merchant purchases a gadget for $30$ less $15\%$. He aims to sell the gadget at a gain of $25\%$ on his cost after allowing a $10\%$ discount on his marked price. At what price, in dollars, should the gadget be marked?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35.42}
Compute the intersection number of the pullbacks of the line bundles \(\iota^*\mathcal{O}(2,3)\) and \(\iota^*\mathcal{O}(2,3)\) on the smooth subvariety \(X = V_+(f) \subset \mathbb{P}^2 \times \mathbb{P}^1\), where \(f \in H^0(\mathcal{O}(1,2))\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
A sequence \( i_{1}, i_{2}, \ldots, i_{n} \) is formed from the first \( n > 3 \) natural numbers, arranged in random order. What is the probability that for all \( k = 1, 2, \ldots, n \), the inequality \( i_{k} \geq k - 3 \) holds true?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4^{n-3} \cdot 6}{n!}}
Evaluate the limit \( \lim_{\epsilon\to 0^+} \frac{3}{4\pi\epsilon^3}\oint_{S_\epsilon} G \cdot\hat{N}dS \), where \( S_\epsilon \) is the sphere centered at \((a,b,c)\) with radius \( \epsilon \), and the vector field is given by \( G(x,y,z) = (x^2y, z^2x, y^2z) \). Here, \( \hat{N} \) is the outward unit normal vector to the sphere.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2ab + b^2}
If the circle $(x+2)^2 + (y-a)^2 = 1$ intersects with the circle $(x-a)^2 + (y-5)^2 = 16$, the range of the real number $a$ is _________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \in (1, 2)}
Calculate the probability that a random variable \( \xi \) from a Beta distribution with parameters \((1,2)\) is less than a random variable \( \eta \) from an Exponential distribution with parameter \(1\), given that \( \xi \) and \( \eta \) are independent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{e}}
A drink vendor has some liters of Maaza, 144 liters of Pepsi and 368 liters of Sprite. He wants to pack them in cans, so that each can contains the same number of liters of a drink, and doesn't want to mix any two drinks in a can. The least number of cans required is 143. How many liters of Maaza does the vendor have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1776}
Find the dimension of the quotient ring \( \mathbb{K}[x,y,z]/(xy,xz,yz) \) where \( \mathbb{K} \) is a field.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
正整数 $9+95+995+\cdots+\underbrace{99 \cdots 9}_{2016} 5$ 的十进制表示中数值 1 的个数为多少？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2014}
The decreasing sequence $a, b, c$ is a geometric progression, and the sequence $19a, \frac{124b}{13}, \frac{c}{13}$ is an arithmetic progression. Find the common ratio of the geometric progression.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{247}
Given the sequence $\left\{a_{n}\right\}$ satisfying:
$$
a_{0}=\sqrt{6}, \quad a_{n+1}=\left[a_{n}\right]+\frac{1}{\left\{a_{n}\right\}} \text {, }
$$

where $[a]$ denotes the greatest integer less than or equal to the real number $a$, and $\{a\}=a-[a]$. Determine the value of $a_{2020}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6060 + \sqrt{6}}
If \(\sin \left(c^{2}-3 c+17\right)^{\circ}=\frac{4}{b-2}\), where \(0 < c^{2}-3 c +17 < 90\) and \(c > 0\), find the value of \(c\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
In this version of SHORT BINGO, a $5\times5$ card is again filled by marking the middle square as WILD and placing 24 other numbers in the remaining 24 squares. Now, the card is made by placing 5 distinct numbers from the set $1-15$ in the first column, 5 distinct numbers from $11-25$ in the second column, 4 distinct numbers from $21-35$ in the third column (skipping the WILD square in the middle), 5 distinct numbers from $31-45$ in the fourth column, and 5 distinct numbers from $41-55$ in the last column. How many distinct possibilities are there for the values in the first column of this SHORT BINGO card?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{360360}
A square flag features a symmetric red cross with uniform width and two identical blue squares in the center on a white background. The entire cross, including the red arms and the two blue centers, occupies 40% of the flag's area. Determine what percent of the flag's area is occupied by the two blue squares.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20\%}
In the year 1990, there were 5000 men, 3000 women, and 2000 boys. In 1994, the number of men increased by a certain percentage and the number of women increased by the ratio of boys to women. The total population in 1994 was 13000. What was the percentage increase in the number of men?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20\%}
If you multiply two integers together and then add 4, the result is 40. What sum of the two numbers could not be possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
A sequence $a_0,a_1,a_2,\cdots,a_n,\cdots$ satisfies that $a_0=3$, and $(3-a_{n-1})(6+a_n)=18$, then the value of $\sum_{i=0}^{n}\frac{1}{a_i}$ is________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2^{n+2} - n - 3}{3}}
The mathematician John is having trouble remembering his girlfriend Alicia's 7-digit phone number. He remembers that the first four digits consist of one 1, one 2, and two 3s. He also remembers that the fifth digit is either a 4 or 5. While he has no memory of the sixth digit, he remembers that the seventh digit is 9 minus the sixth digit. If this is all the information he has, how many phone numbers does he have to try if he is to make sure he dials the correct number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{240}
Evaluate the sum \( \sum_{n=0}^\infty \frac{1}{n!(n^4+n^2+1)}. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{e}{2}}
Determine the number of possible bases for a field extension $M/K$ where $|M|=p^3$ and $K=\mathbb{F}_{p}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(p^3-1)(p^3-p)(p^3-p^2)}
Calculate the least number of intervals \(n\) required to approximate the definite integral of \(5e^{x^2}\) from 0 to 1 using the Error Bound formula with an error less than \(10^{-9}\). The given error bound formula is:
\[Error(S_n) \le \frac{k(b-a)^5}{180N^4}\]
where \(f^{(4)}(x) = 20e^{x^2}(4x^4 + 12x^2 + 13)\), and the maximum of \(f^{(4)}(x)\) occurs at \(x = 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{306}
A coffee express machine has the following probabilities for each request: it brews coffee with probability \( p_c = 0.9 \), pours hot water with probability \( p_w = 0.09 \), and gets out of order with probability \( p_f = 0.01 \). Let \( X \) denote the number of coffees the machine prepares before it gets out of order. Compute \( \operatorname{E}[X] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
Determine if there exists a distinct non-zero integer $k$ such that the equation $x^4 + y^4 = k$ has more than one solution in terms of distinct non-zero integer pairs $(x, y)$. If such a $k$ exists, provide an example.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{635318657}
How many unique $3$-letter sequences with no spaces can be made using the letters in "AUGUSTIN LOUIS CAUCHY", which contains $19$ letters? For example, "GAA" is one acceptable sequence, but "GGA" is not an acceptable sequence because there is only one G available. The original ordering of the letters does not have to be preserved.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1486}
In a 3-dimensional space, there are 2016 spheres, none of which coincide. Some of the spheres are red, and the others are green. Each contact point between a red sphere and a green sphere is painted blue. Find the maximum possible number of blue points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1016064}
Two vertical towers, \( AB \) and \( CD \), are located \( 16 \mathrm{~m} \) apart on flat ground. Tower \( AB \) is \( 18 \mathrm{~m} \) tall and tower \( CD \) is \( 30 \mathrm{~m} \) tall. Ropes are tied from \( A \) to \( C \) and from \( B \) to \( C \). Assuming the ropes are taut, calculate the total length of rope in meters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
Calculate the residue of the function \( f(z)=\frac{1}{1+9z^2} \) at the pole \( z=\frac{i}{3} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{i}{6}}
How many positive integers $N$ satisfy all of the following three conditions?

1. $N$ is divisible by $2020$.
2. $N$ has at most $2020$ decimal digits.
3. The decimal digits of $N$ are a string of consecutive ones followed by a string of consecutive zeros.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{508536}
Charles bought 20 papers to draw, consisting of 10 regular papers and 10 watercolor papers. Today, Charles drew 6 pictures, 4 on regular papers, and 2 on watercolor papers. Yesterday, he drew 6 pictures before going to work, all of which were on regular papers. He also committed to using only watercolor papers for all the drawings when he came back from work yesterday. If he has 2 regular papers left, how many watercolor pictures did Charles draw after coming back from work yesterday, given that at least one of the remaining drawings must be on a watercolor paper?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
You can lower the price by 20% if you buy more than fifteen units of iPhone cases. If you pay $500 to buy 18 units, what is the original price?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{625}
Suppose \( f(x+1) = 2f(x) \) for all \( x \in \mathbb{R} \), and \( f(x) = x(x-1) \) for \( 0 < x \leq 1 \). Find the value of \( m \) such that \( f(x) \geq -\frac{8}{9} \) for all \( x \leq m \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{3}}
There is an unlimited number of chips in six colors. What is the minimum number of chips that need to be arranged in a row so that for any two different colors in the row, there are two adjacent chips of those colors?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
A safe has three locks, each with 8 possible settings (1 through 8). The safe opens if any two of the three locks are set correctly. What is the minimum number of combinations needed to guarantee opening the safe?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
已知函数$f(x)=\sin (\omega x+\frac{\pi }{6})+\frac{1}{2}(\omega >0)$，点$P$，$Q$，$R$是直线$y=m(m>0)$与函数$f(x)$的图象自左至右的某三个相邻交点，且$2|PQ|=|QR|=\frac{3\pi }{2}$，则$\omega +m=$____．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{17}{9}$}
Given acute angles $α$ and $β$ that satisfy: $\cos α= \frac {1}{3}$ and $\cos (α+β)=- \frac {1}{3}$, find $\cos (α-β)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{23}{27}}
Let $A B C$ be an acute triangle with $A$-excircle $\Gamma$. Let the line through $A$ perpendicular to $B C$ intersect $B C$ at $D$ and intersect $\Gamma$ at $E$ and $F$. Suppose that $A D=D E=E F$. If the maximum value of $\sin B$ can be expressed as $\frac{\sqrt{a}+\sqrt{b}}{c}$ for positive integers $a, b$, and $c$, compute the minimum possible value of $a+b+c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{705}
Suppose  $A=\{1,2,\dots,2002\}$  and  $M=\{1001,2003,3005\}$ .  $B$  is an non-empty subset of  $A$ .  $B$  is called a  $M$ -free set if the sum of any two numbers in  $B$  does not belong to  $M$ . If  $A=A_1\cup A_2$ ,  $A_1\cap A_2=\emptyset$  and  $A_1,A_2$  are  $M$ -free sets, we call the ordered pair  $(A_1,A_2)$  a  $M$ -partition of  $A$ . Find the number of  $M$ -partitions of  $A$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{501}}