inclusionAI/Ring-lite-rl-data huggingface dataset:
Suppose a sequence $\{a\_n\}$ satisfies $\frac{1}{a\_{n+1}} - \frac{1}{a\_n} = d (n \in \mathbb{N}^*, d$ is a constant), then the sequence $\{a\_n\}$ is called a "harmonic sequence". It is known that the sequence $\{\frac{1}{x\_n}\}$ is a "harmonic sequence", and $x\_1 + x\_2 + ... + x\_{20} = 200$, find the maximum value of $x\_3 x\_{18}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
For each positive integer, define a function \[ f(n)=\begin{cases}0, &\text{if n is the square of an integer}  \left\lfloor\frac{1}{\{\sqrt{n}\}}\right\rfloor, &\text{if n is not the square of an integer}\end{cases}. \] Find the value of  $\sum_{k=1}^{200} f(k)$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{629}
Let $$g(x)= \frac{1}{2}\sum_{k=0}^{\infty} \frac{(2x)^{2k+1}}{(2k+1){2k\choose k}}$$

If the value of $g(\frac{\sqrt{2}}{2})$ is $\dfrac{\pi}{a}$, where $a\in\Bbb R$, then what is the value of $a$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{2}}
Let $f(x)=c x(x-1)$, where $c$ is a positive real number. We use $f^{n}(x)$ to denote the polynomial obtained by composing $f$ with itself $n$ times. For every positive integer $n$, all the roots of $f^{n}(x)$ are real. What is the smallest possible value of $c$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Suppose that \( \mathbf{G} = 10^{100} \). (\( \mathbf{G} \) is known as a googol.) How many times does the digit 9 occur in the integer equal to \( \mathbf{G} - 1009^{2} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96}
At 12:00 hours, J starts to walk from his house at 6 kmph. At 13:30, P follows him from J's house on his bicycle at a certain speed. J will be 3 km behind P at 19.3 hours. What is P's cycling speed in kmph?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
A unit equilateral triangle is given. Divide each side into three equal parts. Remove the equilateral triangles whose bases are the middle one-third segments. Now we have a new polygon. Remove the equilateral triangles whose bases are the middle one-third segments of the sides of the polygon. After repeating these steps infinitely, find the area of the new shape. The original answer is in \(\dfrac{\sqrt{k}}{m}\) format, please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Given the parabola $y^{2}=2px (p > 0)$ and the circle $x^{2}+y^{2}-4x-5=0$, find the value of $p$ if the parabola's axis is tangent to the circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
An even power of a certain number is a four-digit number with the first digit as 3 and the last digit as 5. Find this number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55}
Compute the sum of the positive integers $n \le 100$ for which the polynomial $x^n + x + 1$ can be written as the product of at least 2 polynomials of positive degree with integer coefficients.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1648}
Given that the points (2, 1) and (-1, 3) are on opposite sides of the line $3x-2y+a=0$, find the range of values for $a$. Express your answer as an interval in the format $\boxed{[lower, upper]}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-4, 9)}
Given the hyperbola $x^{2}- \frac {y^{2}}{m}=1$ with its left and right foci being $F_{1}$ and $F_{2}$ respectively, a line passing through point $F_{2}$ intersects the right branch of the hyperbola at points $A$ and $B$. If $\triangle ABF_{1}$ is an isosceles triangle with $A$ as the vertex of the right angle, then the value of the real number $m$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4-2 \sqrt {2}}
Given that \(F_{1}\) and \(F_{2}\) are the foci of the ellipse \(\frac{x^{2}}{4} + y^{2} = 1\), and \(A\) and \(B\) are the left vertex and the top vertex of this ellipse, respectively. Point \(P\) is on line segment \(AB\). Determine the minimum value of \(\overrightarrow{P F_{1}} \cdot \overrightarrow{P F_{2}}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{11}{5}}
Given a positive integer \( n \geq 3 \), for an \( n \)-element real array \(\left(x_{1}, x_{2}, \cdots, x_{n}\right)\), if every permutation \( y_{1}, y_{2}, \cdots, y_{n} \) of it satisfies \(\sum_{i=1}^{n-1} y_{i} y_{i+1} \geq -1\), then the real array \(\left(x_{1}, x_{2}, \cdots, x_{n}\right)\) is called "glowing". Find the largest constant \( C = C(n) \) such that for every glowing \( n \)-element real array, \(\sum_{1 \leq i < j \leq n} x_{i} x_{j} \geq C \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
A string has 150 beads of red, blue, and green colors. It is known that among any six consecutive beads, there is at least one green bead, and among any eleven consecutive beads, there is at least one blue bead. What is the maximum number of red beads that can be on the string?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{112}
Using the digits $0,1,2,3, \cdots, 9$ exactly once, find the largest possible multiple of 12.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9876543120}
Let $\mathcal{T}$ be a set of all linear maps from $\mathbf{L\,}(\mathbb{R^2})$ such that the following equality holds:

$$\mathbf{A^{4}-5A^2+4I=0}$$

Find the maximum number of elements in $\mathcal{T}$ such that no two maps are similar.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
A train 250 m long running at a certain speed crosses a platform in 15 sec. The length of the platform is 50.024 m. What is the speed of the train in kmph?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72.006}
Find a complex number $z$ such that $z^{2}+3$ is purely imaginary. Then $z=$____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1+2i}
How many of the numbers $1 \cdot 2 \cdot 3$, $2 \cdot 3 \cdot 4$, ..., $2020 \cdot 2021 \cdot 2022$ are divisible by $2020$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Let \omega=\cos \frac{2 \pi}{727}+i \sin \frac{2 \pi}{727}$. The imaginary part of the complex number $$\prod_{k=8}^{13}\left(1+\omega^{3^{k-1}}+\omega^{2 \cdot 3^{k-1}}\right)$$ is equal to $\sin \alpha$ for some angle $\alpha$ between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$, inclusive. Find $\alpha$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{12 \pi}{727}}
Peter received a new jigsaw with a function for measuring the length of cuts for his birthday. To test the gift, Peter took a square piece of plywood with a side of 50 cm, and cut it into squares with sides of 10 cm and 20 cm. How many squares did he obtain in total if the jigsaw displayed a total cut length of \(2 \mathbf{m} 80 \mathrm{~cm}\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Given the function $f(x)=4\sin (2x+\frac{\pi }{6})$ where $0\leqslant x\leqslant \frac{91\pi }{6}$, let ${x\_1}, {x\_2}, \cdots , {x\_n}$ denote all the zeros of the function $F(x)=f(x)-3$ in ascending order, i.e., ${x\_1} < {x\_2} < \cdots < {x\_n}$. Find the value of ${x\_1}+2{x\_2}+2{x\_3}+\cdots +2{x_{n-1}}+{x\_n}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{445\pi}
A $4 \times 4$ window is made out of 16 square windowpanes. How many ways are there to stain each of the windowpanes, red, pink, or magenta, such that each windowpane is the same color as exactly two of its neighbors?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
In a plane rectangular coordinate system, if the equation \( m (x^2 + y^2 + 2y + 1) = (x - 2y + 3)^2 \) represents an ellipse, what is the range of values for \( m \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(5,+\infty)}
The current birth rate per certain number of people is 52, whereas the corresponding death rate is unknown per the same number of people. If the net growth rate in terms of population increase is 1.2 percent, and there are initially 3000 persons, what is the death rate per certain number of people?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
You have 2020 piles of coins in front of you. The first pile contains 1 coin, the second pile contains 2 coins, the third pile contains 3 coins, and so on. Therefore, the 2020th pile contains 2020 coins. Guess a positive integer $k$, such that the piles contain at least $k$ coins. Take away exactly $k$ coins from these piles. Find the minimum number of turns you need to take away all of these coins.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Problem 7.2. (15 points) Given an isosceles triangle $A B C(A B=B C)$. On the sides $A B, B C, A C$ points $K, L, M$ are marked respectively such that $\angle A K M=90^{\circ}, \angle B L K=90^{\circ}$ and $K M=K L$. What is the measure of angle $C M L ?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
Three people, including one girl, are to be selected from a group of $3$ boys and $2$ girls, determine the probability that the remaining two selected individuals are boys.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{3}}
Let $a_1, a_2, a_3, \dots$ be a sequence of positive real numbers such that $a_k a_{k+2} = a_{k+1} + 1$ for all positive integers $k$. If $a_1$ and $a_2$ are positive integers, find the maximum possible value of $a_{2014}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given a natural number $n$, consider the function $$ f_n(x, y) = x^n + x^{n−1}y + x^{n−2}y^2 + \dots + xy^{n−1} + y^n $$ of two real variables. Find the minimal number $k$ for which there exist functions $g_1, \dots , g_k, h_1, \dots , h_k$ of one real variable such that $$ f_n(x, y) = \sum_{i=1}^k g_i(x)h_i(y) $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n+1}
Given positive integers \(a_{1}, a_{2}, \cdots, a_{2018}\) satisfying
\[ a_{1} < a_{2} < \cdots < a_{2018}, \]

for \(i = 1, 2, \cdots, 2018\), let \(b_{i}\) be the number of positive integers among \(a_{1}, a_{2}, \cdots, a_{2018}\) that are less than or equal to \(i\). Then, find the value of
\[ \frac{\sum_{k=1}^{2018} a_{k} + \sum_{k=1}^{a_{2018}} b_{k}}{a_{2018} + 1}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2018}
Given that |m| = 3, |n| = 2, and mn < 0, calculate the value of m + n. Express your answer as ±a, where a is an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±1}
Given proposition $p: \forall x\geqslant 0$, $x^{2}-x\geqslant 0$, what is its negation, $\neg p$? Express your answer as a logical statement in the format of a proposition.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\exists x\geqslant 0, x^{2}-x < 0}
the bankers discount of a certain sum of money is rs . 72 and the true discount on the same sum for the same time is rs . 60 . the sum due is :
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{300}
A regular $\triangle ABC$ has each of its sides divided into four equal parts. Through each division point, lines parallel to the other two sides are drawn. These lines intersect with the sides and the other parallel lines, creating 15 lattice points. If $n$ points are chosen from these 15 lattice points, determine the minimum value of $n$ such that it is guaranteed that three of the chosen points will form an isosceles triangle (including an equilateral triangle). Calculate the minimum value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Farmer Yang has a \(2015 \times 2015\) square grid of corn plants. One day, the plant in the very center of the grid becomes diseased. Every day, every plant adjacent to a diseased plant becomes diseased. After how many days will all of Yang's corn plants be diseased?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2014}
Example 1.12 A and B are playing table tennis, and the final score is $20: 20$. How many scenarios are there in which A has been leading B throughout the game?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{38!}{20!19!}}
The product of nine consecutive natural numbers is divisible by 1111. What is the smallest possible value that the arithmetic mean of these nine numbers can take?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{97}
The school is having a book-a-thon. The winning class gets a pizza party. The fifth grade has 20 students and one week to read as much as possible. The 6th grade already finished and read a total of 299 hours. How many hours does each student in 5th grade need to average per day to beat them by 1?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given that $F$ is the right focus of the hyperbola $C$: ${{x}^{2}}-\dfrac{{{y}^{2}}}{8}=1$, and $P$ is a point on the left branch of $C$, $A(0,4)$. When the perimeter of $\Delta APF$ is minimized, the area of this triangle is \_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{36}{7}}
An urn is filled with coins and beads, all of which are either silver or gold. Twenty percent of the objects in the urn are beads. Forty percent of the coins in the urn are silver. What percent of objects in the urn are gold coins?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48\%}
Find the smallest natural number that leaves a remainder of 2 when divided by 3, 4, 6, and 8.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
Find the equation of the asymptote of the graph of $r = \cos 2 \theta \sec \theta.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x = -1}
Two tribes share one shaman. One tribe lives in the river valley, and the other on a hillside. Due to the spread of the internet and rumors, the shaman had to lower the prices of his services. For the tribe living in the valley, prices decreased by $10\%$ "in furs" and $15\%$ "in fish". The ratio of fur to fish prices on the local market remains constant. For the tribe on the hillside, the price "in furs" decreased by $20\%$. By how many percent did the price "in fish" decrease for this tribe? Round the answer to one decimal place.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24.4\%}
Given vectors a=(8, 1/2x) and b=(x, 1), where x > 0, if (a-2b) || (2a+b), calculate the value of x.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In triangle ABC, the sides opposite to angles A, B, and C are a, b, and c, respectively. Let S be the area of triangle ABC. If 3a² = 2b² + c², find the maximum value of $\frac{S}{b^{2}+2c^{2}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{14}}{24}}
How do you determine the expected number of tests needed to fix an electronic circuit with $X$ components, where the circuit functions only if all components are working, and $Y$ components are initially defective? Given that a random component is tested and replaced if defective, and the process continues until the circuit works, find the expected number of tests in the limit as both $X$ and $Y$ become large. Express your answer in terms of $X$ and Euler's constant $\gamma$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{X(\ln Y + \gamma)}
Given the curve 
\[
(x - \arcsin \alpha)(x - \arccos \alpha) + (y - \arcsin \alpha)(y + \arccos \alpha) = 0 
\]
is intersected by the line \( x = \frac{\pi}{4} \), determine the minimum value of the length of the chord intercepted as \( \alpha \) varies.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{2}}
Quadrilateral $ABCD$ has right angles at $B$ and $D$, and $AC=3$. If $ABCD$ has two sides with distinct integer lengths, then what is the area of $ABCD$? Express your answer in simplest radical form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt 2+\sqrt 5}
In a village in Sri Lanka, 10% of the population died due to bombardment. Of the remaining population, 20% left the village on account of fear. After some time, 5% of those who left returned to the village, and an additional 8% of the current population were injured and hospitalized. If the total population, including those hospitalized, is now 3,240, how many people were in the village in the beginning?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4,115}
At the cinema, guests were seated in rows. If there are 30 chairs between the guest sitting on the far left and the guest sitting on the far right, how many chairs are occupied with guests?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Let the sequence $\{a_n\}$ have a sum of the first $n$ terms denoted by $S_n$, and given $a_1=1$, $a_n+a_{n+1}= \frac{1}{2^n}$ $(n=1,2,3,\ldots)$, then $S_{2n+3}=$ ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{3}\left(1- \frac{1}{{4}^{n+2}}\right)}
Let $P$ be the product of all non-zero digits of the positive integer $n$. For example, $P(4) = 4$, $P(50) = 5$, $P(123) = 6$, $P(2009) = 18$.  
Find the value of the sum: $P(1) + P(2) + \ldots + P(2008) + P(2009)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{194763}
17. Determine the number of ways of tiling a $4 \times 9$ rectangle by tiles of size $1 \times 2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6336}
Find the smallest positive integer $b$ for which $x^2 + bx + 1764$ factors into a product of two polynomials, each having integer coefficients.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
We color a $98 \times 98$ checkerboard in the style of a chessboard. A move consists of selecting a rectangle made up of small squares and inverting their colors. What is the minimum number of moves required to make the entire checkerboard monochromatic?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{98}
Alex has 2 cakes - the first cake is cut into 8 slices and the second cake is cut into 12 slices. Alex gives away a fourth of the slices from the first cake and a third of the slices from the second cake to his friends. After that, he gives half of the remaining slices from both cakes to his family. Alex eats 4 slices from the first cake and 2 slices from the second cake. How many slices of the cake are left?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
If a, b are two positive, co-prime integers such that $$\lim _{n \rightarrow \infty}(\frac{^{3n}C_n}{^{2n}C_n})^\frac{1}{n}=\frac{a}{b}$$ then what is the value of a + b?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43}
On the extension of side $AD$ of rectangle $ABCD$ beyond point $D$, point $E$ is taken such that $DE = 0.5 AD$ and $\angle BEC = 30^\circ$. 

Find the ratio of the sides of rectangle $ABCD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}:2}
Suppose that $a$, $b$, $c$ and $d$ are positive integers satisfying all of the following relations. \[abcd=2^6\cdot 3^9\cdot 5^7\] \[\text{lcm}(a,b)=2^3\cdot 3^2\cdot 5^3\] \[\text{lcm}(a,c)=2^3\cdot 3^3\cdot 5^3\] \[\text{lcm}(a,d)=2^3\cdot 3^3\cdot 5^3\] \[\text{lcm}(b,c)=2^1\cdot 3^3\cdot 5^2\] \[\text{lcm}(b,d)=2^2\cdot 3^3\cdot 5^2\] \[\text{lcm}(c,d)=2^2\cdot 3^3\cdot 5^2\] What is $\text{gcd}(a,b,c,d)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$3$}
If the equation $a^x = 3m - 2$ ($a > 0$ and $a \neq 1$) has a negative root, find the range of the real number $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, +\infty)}
Anthony writes the  $(n+1)^2$  distinct positive integer divisors of  $10^n$ , each once, on a whiteboard. On a move, he may choose any two distinct numbers  $a$  and  $b$  on the board, erase them both, and write  $\gcd(a, b)$  twice. Anthony keeps making moves until all of the numbers on the board are the same. Find the minimum possible number of moves Anthony could have made.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^2 + n}
Given triangle $ABC$. Let $A_1B_1$, $A_2B_2$, $\ldots$, $A_{2008}B_{2008}$ be $2008$ lines parallel to $AB$ which divide triangle $ABC$ into $2009$ equal areas. Calculate the value of $$ \left\lfloor \frac{A_1B_1}{2A_2B_2} + \frac{A_1B_1}{2A_3B_3} + \ldots + \frac{A_1B_1}{2A_{2008}B_{2008}} \right\rfloor$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43}
Each of two boxes contains both black and white marbles, and the total number of marbles in the two boxes is $25.$ One marble is taken out of each box randomly. The probability that both marbles are black is $27/50,$ and the probability that both marbles are white is $m/n,$ where $m$ and $n$ are relatively prime positive integers. What is $m + n$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
Two individuals, A and B, participate in an exam that consists of a pool of 10 questions. A can answer 6 of these questions correctly, while B can answer 8 correctly. For each test, 3 questions are randomly selected from the pool. The passing criterion is to answer at least 2 questions correctly. Calculate the probability that at least one of A or B passes the test. Express your answer as a fraction in simplest form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{44}{45}}
Four lighthouses are located at points $A$, $B$, $C$, and $D$. The lighthouse at $A$ is $5$ kilometers from the lighthouse at $B$, the lighthouse at $B$ is $12$ kilometers from the lighthouse at $C$, and the lighthouse at $A$ is $13$ kilometers from the lighthouse at $C$. To an observer at $A$, the angle determined by the lights at $B$ and $D$ and the angle determined by the lights at $C$ and $D$ are equal. To an observer at $C$, the angle determined by the lights at $A$ and $B$ and the angle determined by the lights at $D$ and $B$ are equal. The number of kilometers from $A$ to $D$ is given by $\frac{p\sqrt{q}}{r}$, where $p$, $q$, and $r$ are relatively prime positive integers, and $r$ is not divisible by the square of any prime. Find $p + q + r$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96}
Consider the function $$f(x, y, z, u) = \frac{x(1-x)y(1-y)z(1-u)}{1 - (1 - xy)z}$$ for $x, y, z, u \in (0,1)$. Find the supremum of $f(x, y, z, u)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given a sequence ${a_n}$ where the sum of the first $n$ terms is $S_n$, $a_1=1$, and it satisfies $S_n=a_{n+1}$, find the sum of the first 10 terms of the sequence ${S_n}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{512}