inclusionAI/Ring-lite-rl-data huggingface dataset:
Three circles $C_{1}$, $C_{2}$, and $C_{3}$ with radii 1, 2, and 3, respectively, touch each other externally. The centers of $C_{1}$ and $C_{2}$ are on the x-axis, and $C_{3}$ touches them from above. Determine the radius of the circle that is enclosed by $C_{1}$, $C_{2}$, and $C_{3}$ and is tangent to all three circles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{6}{23}}
Evaluate \( \lim_{n\to \infty} \int_{n^2+1}^{n^2+n+1}\frac{\sin^{-1}x}{\sqrt{x}}\,dx \) without performing the integration.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given that the sequence $\{a\_n\}$ is an arithmetic progression, and the sequence $\{b\_n\}$ satisfies $b\_n=a\_n a_{n+1} \cdot a_{n+2} (n \in \mathbb{N}^*)$, let $S\_n$ be the sum of the first $n$ terms of $\{b\_n\}$. If $a_{12}=\frac{3}{8} a_{5} > 0$, find the value of $n$ when $S\_n$ reaches its maximum.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
A system consists of 7 components with independent, identically distributed lifetimes. The system will operate until any one of the components fails. If the lifetime \( X \) of each component has a probability density function given by:

\[f(x) = \begin{cases} \frac{3}{x^4}, & \text{for } x > 1\\ 0, & \text{otherwise} \end{cases}\]

Calculate the expected lifetime until failure of the system.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{21}{20}}
Find the minimum distance from the point \( P = (2, 5, 1) \) to the ellipsoid \( 2x^2 + 4y^2 + z^2 = 9 \), restricted to the first octant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.86}
Calculate \(\int_{S}\omega\), where \(\omega\) is the 2-form on \(\mathbb{R}^3-\{(1,0,0),(-1,0,0)\}\) given by:
\[
\omega=((x-1)^2+y^2+z^2)^{-3/2}((x-1)dy\wedge dz+ydz\wedge dx+zdx \wedge dy)+
  ((x+1)^2+y^2+z^2)^{-3/2}((x+1)dy\wedge dz+ydz\wedge dx+zdx \wedge dy)
\]
and \(S=\{(x,y,z)\in \mathbb{R}^3: x^2+y^2+z^2=5 \}\). The orientation of \(S\) is the natural orientation induced by \(D=\{(x,y,z)\in \mathbb{R}^3: x^2+y^2+z^2 \leq 5 \}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8\pi}
In a $33 \times 33$ grid, each cell is colored with one of three different colors such that the number of cells of each color is equal. Determine the minimum number of separating edges, where a separating edge is defined as an edge between two adjacent cells of different colors.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{66}
Determine the values of $q$ for which the integral \( \int_1^\infty \frac{x^q}{\ln x}\,dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{q < -1}
Evaluate the limit \( \lim_{R \to \infty} \int_{0}^{\frac{\pi}{2}} Re^{-R\sin\theta} \, d\theta \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Compute the covariance \( \text{Cov}[X,Y] \) for the joint mass density function given by: \[ f_{X,Y}(x,y) = \frac{{e}^{-(x + \frac{y}{x})}}{x}, \ y > 0, \ x > 0. \] Assume that \( X \) follows an exponential distribution with parameter 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the number of distinct sequences of rational numbers that converge to 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Let \( f = \sum_{i \ge m} a_i t^i \) be a formal Laurent series, where \( m \in \mathbb{Z} \). Consider the formal differential \( df \). What is the coefficient of degree \(-1\) in the differential form \( \frac{df}{f} \)? Provide an explicit expression for this coefficient.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m}
Determine the order of the singularity at \( z_0 \) for the function \( \frac{f(z) + g(z)}{r(z) + h(z)} \), where \( f(z) \), \( g(z) \), \( r(z) \), and \( h(z) \) are analytic near \( z_0 \). At \( z_0 \), \( f(z) \) has a pole of order 2, \( g(z) \) has a zero of order 3, \( r(z) \) has a zero of order 2, and \( h(z) \) has a zero of order 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Hong and Song each have a shuffled deck of eight cards, consisting of four red and four black cards. Each turn, they play the top two cards from their decks, which can be either two black cards, two red cards, or one of each color. What is the probability that Hong and Song play exactly the same pairs as each other for all four turns? If this probability is expressed as $\frac{m}{n}$, where $m$ and $n$ are coprime positive integers, compute $100m+n$. \(\text{(Proposed by Sean Li)}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25150}
Calculate the norm \( \|\Lambda\| \) of the linear functional \( \Lambda(f) = \int_{0}^{1} (9t^4 - 18t^3 + 11t^2 - 2t) f(t) \, dt \) in the Banach space \( C([0, 1]) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{49}{810}}
Petya has a deck of 36 cards, consisting of 4 suits with 9 cards each. He selects 18 cards to give to Vasya, keeping the remaining 18 for himself. The players take turns placing one card on the table, starting with Petya. If Vasya can place a card of the same suit or rank in response to Petya's move, he earns 1 point. What is the maximum number of points that Vasya can guarantee?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Evaluate the limit: $$\lim_{n\to \infty} \frac{1}{n}\cdot \left((m+1)(m+2) \ldots(m+n)\right)^{\frac{1}{n}}$$ where $m$ is a fixed positive integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{e}}
Find the matrix that corresponds to a dilation centered at the origin with scale factor $-3.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\begin{pmatrix} -3 & 0 \\ 0 & -3 \end{pmatrix}}
On February 1, it was 16.2°C outside Jacinta's house at 3:00 p.m. On February 2, it was -3.6°C outside Jacinta's house at 2:00 a.m. If the temperature changed at a constant rate between these times, calculate the rate at which the temperature decreased. Express your answer in °C per hour.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.8}
Evaluate the double integral \( \iint_{\Omega} (x+y^2) \, dx \, dy \) where \( \Omega \) is the region bounded by the lines \( x+y=0 \), \( x+y=1 \), \( 2x-y=0 \), and \( 2x-y=3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{22}{27}}
Given seven points on a plane, where the distances between them are denoted by $a_1, a_2, \ldots, a_{21}$, determine the maximum number of times the same distance value can occur among these 21 distances.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Find the minimal distance between the origin and the intersection of the surfaces given by the equations \(x^2 = 2yz\) and \(x^2 + 3y^2 + 2z^2 = 30\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{10}}
Determine the order of the numerical scheme given by \(y_{n+1} = y_n + \frac{h}{2}\left(f(x_n, y_n) + f(x_{n+1}, y_{n+1})\right)\) by inserting the exact solution into the scheme.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Consider a sequence of independent two-dimensional vectors of random variables \((A_n, B_n)_{n=1}^{\infty}\), where each vector is uniformly distributed on the square \([-2,2] \times [-2,2]\). Define \(V_n=(S_n, T_n) = (\sum_{i=1}^n A_i, \sum_{i=1}^n B_i)\) and \(|V_n| = \sqrt{(S_n)^2+(T_n)^2}\). Determine the constant \(c\) such that \(\lim_{n \to \infty} P(|V_n|<c\sqrt{n})=0.95\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.83}
Consider the parabola  $ C: y \equal{} x^2$  in the  $ x \minus{} y$  plane. Let  $ l_1$  be the tangent line of  $ C$  with slope  $ \sqrt {3}$  and  $ l_2$  be the tangent line of  $ C$  with negative slope which makes angle  $ 45^\circ$  with  $ l_1$ . Find the area bounded by  $ C,\ l_1$  and  $ l_2$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5 + 3\sqrt{3}}{6}}
A jar contains 97 marbles that are either red, green, or blue. Neil draws two marbles from the jar without replacement and notes that the probability that they would be the same color is $\frac{5}{12}$. After Neil puts his marbles back, Jerry draws two marbles from the jar with replacement. Compute the probability that the marbles that Jerry draws are the same color.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{41}{97}}
A carton contains milk that is \(2\%\) fat, an amount that is \(40\%\) less fat than the amount contained in a carton of whole milk. What is the percentage of fat in whole milk? Provide your answer as a fraction in the form \(\frac{k}{m}\), where the fraction is in its simplest form. Please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Given vectors OA = a, OB = b, OC = c, and the condition AC = 3BC, express c in terms of a and b. Provide your answer as a vector equation, including all necessary operations and coefficients.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{2}a + \frac{3}{2}b}
Abie initially had 20 bags of chips. She decided to share 2/5 of the bags with her friends and bought 18 bags in the store which only cost her half the price of the initial 20 bags. On her way home, Abie found a coupon to buy 4 more bags of chips for 3/4 of the original price per bag. How much did Abie spend in total if each bag originally cost her $2?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$64}
Let \( f \) be an arbitrary, twice differentiable function for which \( f'' \neq 0 \). The function \( u(x,y) = f(x^2 + axy + y^2) \) satisfies the equation \( U_{xx} - U_{yy} = 0 \). Determine the value of the constant \( a \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm 2}
Determine the cofinality of the ordinal expression \( \omega_3^{\omega_2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\omega_2}
Twenty-five adults went to a soccer match with thirty-two children. Adult tickets are $12 each and the total bill was $450. How many dollars is one child's ticket?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$4.69}
Given two independent standard normal random variables, $X$ and $Z$, find the probability $P(X+Z<0, Z>0)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/8}
Find the sum of all three-digit natural numbers that contain at least one odd digit and at least one even digit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{370775}
Determine the number of subgroups of order 125 in the group $G = \mathbb{Z}_{360} \oplus \mathbb{Z}_{150} \oplus \mathbb{Z}_{75} \oplus \mathbb{Z}_{3}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse $\Gamma$: $\frac{{x}^{2}}{{a}^{2}}+y^{2}=1\left(a \gt 1\right)$, $A$ is the top point of $\Gamma$, and the line $l$ passes through point $F_{1}$ and intersects $\Gamma$ at points $B$ and $C$. If $l$ vertically bisects the line segment $AF_{2}$, then the perimeter of $\triangle ABC$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{8\sqrt{3}}{3}}
Define the sequence \(\{a_n\}\) where \(a_n = n^3 + 4\) for \(n \in \mathbf{N}_+\). Let \(d_n = \gcd(a_n, a_{n+1})\), which is the greatest common divisor of \(a_n\) and \(a_{n+1}\). Find the maximum value of \(d_n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{433}
Given a $4 \times 4$ squares table, how many ways can we fill the table with $\{0,1\}$ such that any two neighboring squares (sharing one common side) have a product equal to $0$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1234}
What is the smallest number of regular hexagons of side length 1 needed to completely cover a disc of radius 1 ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given that \( F_{1} \) and \( F_{2} \) are the left and right foci of the ellipse \(\frac{x^{2}}{4}+\frac{y^{2}}{3}=1\), and \( A \) is a moving point on the ellipse. Circle \( C \) is tangent to the extended line of \( F_{1}A \), the extended line of \( F_{1}F_{2} \), and segment \( AF_{2} \). If \( M(t, 0) \) is a point of tangency, then \( t=\_\_\_\_\_ \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In the "Happy Dictionary," there is a question that presents a sequence of numbers and asks you to fill in the next few numbers based on the pattern. Here is the sequence given: $$- \frac {1}{2}, \frac {1}{2}, - \frac {3}{8}, \frac {1}{4}, \ldots, - \frac {5}{32}, \frac {3}{32}, \ldots$$ The 8th number in the sequence can be \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac {1}{32}}
A river flows at a constant speed. Piers A and B are located upstream and downstream respectively, with a distance of 200 kilometers between them. Two boats, A and B, depart simultaneously from piers A and B, traveling towards each other. After meeting, they continue to their respective destinations, immediately return, and meet again for the second time. If the time interval between the two meetings is 4 hours, and the still water speeds of boats A and B are 36 km/h and 64 km/h respectively, what is the speed of the current in km/h?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Let $a_n$ be the product of the complex roots of $x^{2n} = 1$ that are in the first quadrant of the complex plane. That is, roots of the form $a + bi$ where $a, b > 0$. Let $r = a_1 \cdot a_2 \cdot \ldots \cdot a_{10}$. Find the smallest integer $k$ such that $r$ is a root of $x^k = 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1260}
What is the probability that among the five drawn lottery numbers there are consecutive numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.2075}
In an examination, a student scores 5 marks for every correct answer and loses 2 marks for every wrong answer. If he attempts all 150 questions and secures 370 marks, the number of questions he attempts correctly is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{95}
Josh had 16 marbles in his collection. He decided to triple his collection and then added 10 more marbles. After that, he lost 25% of his marbles and found 5 marbles the next day. What is the square root of the total number of marbles Josh has now?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Find the smallest positive integer  $k$  such that, for any subset  $A$  of  $S=\{1,2,\ldots,2012\}$  with  $|A|=k$ , there exist three elements  $x,y,z$  in  $A$  such that  $x=a+b$ ,  $y=b+c$ ,  $z=c+a$ , where  $a,b,c$  are in  $S$  and are distinct integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1008}
The integers  $1, 2, \dots, 64$  are written in the squares of a  $8 \times 8$  chess board, such that for each  $1 \le i < 64$ , the numbers  $i$  and  $i+1$  are in squares that share an edge. What is the largest possible sum that can appear along one of the diagonals?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{456}
Three of the edges of a [[cube]] are $\overline{AB}, \overline{BC},$ and $\overline{CD},$ and $\overline{AD}$ is an interior [[diagonal]].  Points $P, Q,$ and $R$ are on $\overline{AB}, \overline{BC},$ and $\overline{CD},$ respectively, so that $AP = 5, PB = 15, BQ = 15,$ and $CR = 10.$  What is the [[area]] of the [[polygon]] that is the [[intersection]] of [[plane]] $PQR$ and the cube?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{525}
What is the minimum number of straight cuts required to divide a $5 \times 5$ square into unit squares, given that after each cut the resulting pieces can be rearranged arbitrarily before making the next cut, allowing multiple pieces to be cut simultaneously?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
What is the expected number of random integers in the range [1, N] that need to be generated before encountering a duplicate integer, assuming the generator is ideal?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\sqrt{\pi N/2}$}
Sixteen wooden Cs are placed in a 4-by-4 grid, all with the same orientation, and each is to be colored either red or blue. A quadrant operation on the grid consists of choosing one of the four two-by-two subgrids of Cs found at the corners of the grid and moving each C in the subgrid to the adjacent square in the subgrid that is 90 degrees away in the clockwise direction, without changing the orientation of the C. Given that two colorings are the considered same if and only if one can be obtained from the other by a series of quadrant operations, determine the number of distinct colorings of the Cs.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1296}
$A_1, A_2, ..., A_n$  are the subsets of  $|S|=2019$  such that union of any three of them gives  $S$  but if we combine two of subsets it doesn't give us  $S$ . Find the maximum value of  $n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
A, B, C, and D rent a pasture. If A puts 10 oxen for 7 months, B puts 12 oxen for 5 months, C puts 15 oxen for 3 months, and D puts 20 oxen for 6 months for grazing, and the rent of the pasture is 250 USD. Given the currency conversion rate of 1 USD = 50 local currency units (LCU), how much amount should C pay as his share of rent in LCU?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1907}
Given the function $f(x) = \log_a(8-ax)$ ($a > 0$ and $a \neq 1$), if $f(x) > 1$ holds true in the interval $[1,2]$, then the range of the real number $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1,\dfrac{8}{3})}
In trapezoid $PQRS$ with $PQ$ parallel to $RS$, the diagonals $PR$ and $QS$ intersect at $T$. If the area of triangle $PQT$ is 75 square units, and the area of triangle $PST$ is 30 square units, calculate the area of trapezoid $PQRS$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{147}
The volume of a certain rectangular solid is $8$ cm<sup>3</sup>, its total surface area is $32$ cm<sup>2</sup>, and its three dimensions are in geometric progression. The sums of the lengths in cm of all the edges of this solid is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$32$}
The equation $ ax^3\plus{}bx^2\plus{}cx\plus{}d\equal{}0$ has three distinct solutions. How many distinct solutions does the following equation have:

$ 4(ax^3\plus{}bx^2\plus{}cx\plus{}d)(3ax\plus{}b)\equal{}(3ax^2\plus{}2bx\plus{}c)^2?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Athletes A, B, and C, along with 4 volunteers, are lining up for a group photo. If A and B must be next to each other and C cannot be at either end, find the total number of different ways to arrange them. Express your answer as a numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{960}
In the trapezium FGHI, side GF is parallel to HI. The lengths of FG and HI are 50 and 20 respectively. Point J is on side FG such that segment IJ divides the trapezium into two parts of equal area. Find the length of FJ. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
During training target practice, each soldier fired 10 times. One of them was successful and scored 90 points in total. How many times did he score 7 points if he scored 10 points 4 times and the results of the remaining shots were 7, 8, and 9 points? Note that there were no misses.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
设 $x, y, z \geq 0$, 且至多有一个为 0 , 求

$$
f(x, y, z)=\sqrt{\frac{x^{2}+256 y z}{y^{2}+z^{2}}}+\sqrt{\frac{y^{2}+256 z x}{z^{2}+x^{2}}}+\sqrt{\frac{z^{2}+256 x y}{x^{2}+y^{2}}}
$$

的最小值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$12$}
Given a circle $C: (x-2)^2+y^2=1$ and a point $P$ on the line $l: x+y+1=0$. If there exists a line $m$ passing through point $P$ intersecting circle $C$ at points $A$ and $B$, with point $A$ being the midpoint of $PB$, then the range of the x-coordinate $x_0$ of point $P$ is \_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-1, 2]}
In a tetrahedron \(A B C D\), edge \(A B\) is perpendicular to edge \(C D\), \(\angle A C B = \angle A D B\), the area of the cross-section passing through edge \(A B\) and the midpoint of edge \(D C\) is \(S\), and \(|D C| = a\). Find the volume of tetrahedron \(A B C D\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a S}{3}}
A quality inspector sequentially selects products numbered 3, 13, 23,..., 93 from 100 products numbered 1 to 100 for inspection. What sampling method is used in this case? Provide your answer as the name of the specific sampling method.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Systematic sampling}
In the quadrilateral \(ABCD\), it is known that \(AB = BC\) and \(\angle ABC = \angle ADC = 90^\circ\). A perpendicular \(BH\) is dropped from vertex \(B\) to side \(AD\). Find the area of quadrilateral \(ABCD\) if it is known that \(BH = h\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{h^2}
In trapezoid \(A B C D\), the bases \(A D\) and \(B C\) are 8 and 18, respectively. It is known that the circumscribed circle of triangle \(A B D\) is tangent to lines \(B C\) and \(C D\). Find the perimeter of the trapezoid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
Petya wants to paint several cells of an $8 \times 8$ square such that for any vertex, there is a painted square to which it belongs. What is the minimum number of squares he needs to paint?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Seryozha and Lena have several chocolate bars, each weighing no more than 100 grams. No matter how they divide these chocolate bars, the total weight of the chocolate bars for one of them will not exceed 100 grams. What is the maximum possible total weight of all the chocolate bars?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{300 \text{ grams}}
Determine the number of ways to select a sequence of 8 sets $A_{1}, A_{2}, \ldots, A_{8}$, such that each is a subset (possibly empty) of \{1,2\}, and $A_{m}$ contains $A_{n}$ if $m$ divides $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2025}