inclusionAI/Ring-lite-rl-data huggingface dataset:
Five glass bottles can be recycled to make a new bottle. Additionally, for every 20 new bottles created, a bonus bottle can be made from residual materials. Starting with 625 glass bottles, how many total new bottles can eventually be made from recycling and bonuses? (Keep counting recycled and bonus bottles until no further bottles can be manufactured. Do not include the original 625 bottles in your count.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{163}
Find the largest integer \( m \) such that \( a^{12} \equiv 1 \pmod{m} \) for all integers \( a \) that are relatively prime to \( m \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65520}
A bug travels in the coordinate plane, moving only along the lines that are parallel to the $x$-axis or $y$-axis. Let $A = (-3, 2)$ and $B = (3, -2)$. Consider all possible paths of the bug from $A$ to $B$ of length at most $20$. How many points with integer coordinates lie on at least one of these paths?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$195$}
Evaluate \( \lim_{n \rightarrow \infty} \int_0^1 \frac{nx}{1 + n^2x^4} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
A man travels between three towns forming a triangle by walking, riding, and driving. He can walk, ride, and drive a mile in \(a\), \(b\), and \(c\) minutes respectively. If he starts from town B, he takes \(a + c - b\) hours; if he starts from town C, he takes \(b + a - c\) hours; and if he starts from town A, he takes \(c + b - a\) hours. Determine the total length of the circuit in miles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Determine the condition under which the eigenvalue sets of the matrices \(I - BAA^T\) and \(I - A^TBA\) are equal, where \(A \in \mathbb{R}^{m \times n}\) and \(B \in \mathbb{R}^{m \times m}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m = n}
A man leaves point P at 6 a.m. and reaches point Q at 10 a.m. Another man leaves point Q at 8 a.m. and reaches point P at 12 noon. At what time do they meet?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Given a yearly 3x3 transition matrix with eigenvalues \( \lambda = 1, \frac{1}{2}, 0 \), and corresponding eigenvectors \( x_1 = (60, 20, 30) \) and \( x_2 = (-60, -30, -90) \), the initial state vector is \( v_0 = 500x_1 + 200x_2 + 100x_3 \), where \( x_3 \) is the eigenvector for \( \lambda = 0 \). Determine the number of people who will be sick with the disease 2 years later.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8500}
设 A,B 是集合 \{1,2,\cdots,20\} 的两个子集，A\cap B=\emptyset，且 n\in A 时，2n+2\in B. 记 M(A) 为 A 的元素之和，则 M(A) 的最大值是__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{39}
A side of beef lost 35 percent of its weight in processing. If the side of beef weighed a certain amount after processing, and it weighed 861.54 pounds before processing, how many pounds did it weigh after processing?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{560.001}
Find $1 - 0.\overline{123}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{292}{333}}
4. The numbers $x$ and $y$ are such that the equalities $\cos y + \cos x = \sin 3x$ and $\sin 2y - \sin 2x = \cos 4x - \cos 2x$ hold. What is the smallest value that the sum $\sin y + \sin x$ can take?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1-\frac{\sqrt{2+\sqrt{2}}}{2}}
Determine the value of \( c \) that leads to the fastest convergence of the sequence \( D_n(c) = \frac {4^n}{\binom {2n}{n} \sqrt{4n + c}} \) to \( \frac{\sqrt{\pi}}{2} \) as \( n \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Provide an example of a bounded sequence that is not convergent, but where the differences of consecutive terms converge to zero.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a_n = \sin(\sqrt{n})}
Ying lives on Strangeland, a tiny planet with 4 little cities that are each 100 miles apart from each other. One day, Ying begins driving from her home city of Viavesta to the city of Havennew, which takes her about an hour. When she gets to Havennew, she decides she wants to go straight to another city on Strangeland, so she randomly chooses one of the other three cities (possibly Viavesta), and starts driving there. Ying drives like this for most of the day, making 8 total trips between cities on Strangeland, choosing randomly where to drive to next from each stop. She then stops at her final city of destination, digs a hole, and buries her car.

Let $p$ be the probability Ying buried her car in Viavesta and let q be the probability she buried it in Havennew. Find the value of $p + q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\dfrac{1093}{2187}$}
Let \( f: \mathbb{R} \to \mathbb{R} \) be a function that is three times differentiable and satisfies the equation \( f(x^2)f''(x) = f'(x)f'(x^2) \) for all \( x \in \mathbb{R} \). Given that \( f(1) = 1 \) and \( f'''(1) = 8 \), find the value of \( f'(1) + f''(1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
A point \( P \) inside an equilateral triangle \( ABC \) is at distances \( PA = 6 \), \( PB = 8 \), and \( PC = 10 \) from the vertices. Find the closest integer to the area of \( \triangle ABC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{79}
Given a function $f: \mathbb{N}^+ \rightarrow \mathbb{N}^+$ that is strictly increasing and satisfies $f(f(n)) = 3n$ for all positive integers $n$, find $f(100)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{181}
Determine the distance from the origin to the set \( A \) in \( \mathbb{R}^{2d} \) where \( d > 1 \), defined as:
\[
A = \left\{ \begin{pmatrix} x_1 & y_1 & \ldots & x_d & y_d \end{pmatrix}': x_i^2 + y_i^2 > c, \ i = 1, \ldots, d \right\},
\]
with \( c \) being a positive constant. Calculate \( d(0, A) \), the distance from the origin to the set \( A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{cd}}
Calculate the flux of the vector field \( \bar{F} = \langle x, y, x \rangle \) across the plane defined by \( 1 - (x + y + z) = 0 \) in the first octant, using the unit normal vector \( \langle \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \rangle \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
7. From the sides and diagonals of a regular 12-sided polygon, three different segments are randomly selected. The probability that their lengths are the side lengths of a triangle is $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{223}{286}}
Let \(\kappa\) and \(\lambda\) be infinite cardinals. Consider a random function \(\phi: \kappa \times \lambda \rightarrow \{0,1\}\) constructed by flipping a fair coin for each element of the domain. Determine the probability \(P(\kappa, \lambda)\) that there exists an \(\alpha < \kappa\) such that \(\forall x, \phi(\alpha, x) = 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the total number of different integer values the function $$f(x)=[x]+[2 x]+\left[\frac{5 x}{3}\right]+[3 x]+[4 x]$$ takes for real numbers $x$ with $0 \leq x \leq 100$. Note: $[t]$ is the largest integer that does not exceed $t$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{734}
An article costing $C$ dollars is sold for $100 at a loss of $x$ percent of the selling price. It is then resold at a profit of $x$ percent of the new selling price $S'$. If the difference between $S'$ and $C$ is $1\frac {1}{9}$ dollars, then $x$ is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$10$}
## Task B-3.6.

A rectangle is given with side lengths $a$ and $b$, where $a = \frac{4}{7} b$. Determine the probability that a randomly selected point within the given rectangle is closer to the shorter side and is less than $\frac{a}{8}$ away from it.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{8}}
设在 $5 \times 5$ 的方格表的第 $i$ 行第 $j$ 列所填的数为 $a_{i j}\left(a_{i j} \in\{0,1\}\right), a_{i j}=a_{j i}(1 \leqslant i、j \leqslant 5)$ .则表中共有五个 1 的填表方法总数为 $\qquad$ （用具体数字作答).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{326}
Determine the number of roots of the polynomial \( f(x) = x^{134} + x^{127} + x^7 + 1 \) over the field \( \mathbb{F}_{2^{1463}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{127}
Let $X(t)$ be the price of JetCo stock at time $t$ years from the present. Assume that $X(t)$ follows a geometric Brownian motion with zero drift and volatility $\sigma = 0.4/yr$. If the current price of JetCo stock is 8.00 USD, calculate the probability that the price will be at least 8.40 USD six months from now.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.377}
In a game, 15 players each choose 6 cards from separate decks of 40 cards. Calculate the expected number of cards that at least two players will share.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27.26}
A stick of length 2 meters is made of uniformly dense material. A point is chosen randomly on the stick, and the stick is broken at that point. The left portion of the stick is discarded. This process is repeated indefinitely on the remaining portion of the stick. What is the probability that one of the discarded left parts has a length greater than 1 meter?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln 2}
In a regular quadrilateral pyramid $S A B C D$, where each edge is equal to 2, a section is constructed by a plane parallel to the diagonal of the base $A C$ and the lateral edge $S B$ of the pyramid, intersecting the edge $A B$. Find the perimeter of the polygon obtained in this section, if the lower base of the section is equal to $\sqrt{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2+\sqrt{2}+\sqrt{3}}
On the leg \( BC \) of the right triangle \( ABC \), a circle is constructed with \( BC \) as the diameter. The circle intersects the hypotenuse at point \( D \) such that \( AD : BD = 1 : 3 \). The altitude from vertex \( C \) (the right angle) to the hypotenuse is 3. Find the leg \( BC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given a $7 \times 7$ matrix $A$ such that $2A^2 - A^4 = I$, where $I$ is the identity matrix, and $A$ has two distinct eigenvalues, each with a geometric multiplicity of 3, determine the total number of non-zero entries in the Jordan canonical form of $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
设 V 为关于 x 的次数小于或等于 10 的实多项式构成的集合, 且 a为实数。定义 $V$ 到 V 的线性映射 F 为 $F(f)=\left(x^{2} f\right)^{\prime \prime}+f^{\prime}-a f$ 。记 I 为使得 F 不是满射的 a 的所有可能值之和，则 $[I]=$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{572}
In how many ways can 5 married couples be seated around a circular table such that man 1 is seated next to his wife, and man 3 is not seated next to his wife?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60480}
Example 19 (Problem from the 2nd American Mathematical Invitational Competition) Given
$$
\left\{\begin{array}{l}
\frac{x^{2}}{2^{2}-1^{2}}+\frac{y^{2}}{2^{2}-3^{2}}+\frac{z^{2}}{2^{2}-5^{2}}+\frac{w^{2}}{2^{2}-7^{2}}=1, \\
\frac{x^{2}}{4^{2}-1^{2}}+\frac{y^{2}}{4^{2}-3^{2}}+\frac{z^{2}}{4^{2}-5^{2}}+\frac{w^{2}}{4^{2}-7^{2}}=1, \\
\frac{x^{2}}{6^{2}-1^{2}}+\frac{y^{2}}{6^{2}-3^{2}}+\frac{z^{2}}{6^{2}-5^{2}}+\frac{w^{2}}{6^{2}-7^{2}}=1, \\
\frac{x^{2}}{8^{2}-1^{2}}+\frac{y^{2}}{8^{2}-3^{2}}+\frac{z^{2}}{8^{2}-5^{2}}+\frac{w^{2}}{8^{2}-7^{2}}=1,
\end{array}\right.
$$

Find the value of $x^{2}+y^{2}+z^{2}+w^{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
In the convex quadrilateral ABCD, it is known that ∠DAB = ∠BCD = 90°, and ∠CDA : ∠ABC = 2 : 1, AD : CB = 1 : √3. What is the measure of ∠BDA?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
A nonempty word is called pronounceable if it alternates between vowels (A, E, I, O, U) and consonants (all other letters) and contains at least one vowel. How many pronounceable words can be formed using the letters P, U, M, A, C at most once each? Words of length shorter than $5$ are allowed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{68}
Given that a point $P$ is taken within a right circular cone $(M-ABC)$ such that the volume of the cone $(P-ABC)$ is not greater than one third of the volume of the cone $(M-ABC)$, what is the probability?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{19}{27}}
Find the value of $k$ such that the curves $y=k\sqrt{x^2-1}$ and $y=\ln x$ are tangential.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.402}
If the $60$th percentile of the data $0$, $1$, $2$, $3$, $4$, $5$, $7$, $8$, $9$, $10$ is $n$, find the constant term in the expansion of $(\frac{1}{x}-2x)^n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-160}
Let $\{x_n\}_{n=1}^{\infty}$ be a convergent sequence in a normed space $(S, \|\cdot\|)$. Find the limit of the sequence \[\left\{\frac{x_1 + 2x_2 + \cdots + nx_n}{n^2}\right\}_{n=1}^{\infty}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{L}{2}}
Referees from China, Japan, and Korea, two from each country, will serve as judges for badminton courts 1, 2, and 3 at the Asian Games. It is stipulated that each court will have two referees, and referees from the same country cannot be assigned to the same court. The number of different allocation methods is ____. (Fill in the number)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Petya was running down an escalator, counting the steps. Exactly halfway down, he tripped and tumbled the rest of the way (he tumbles 3 times faster than he runs). How many steps are on the escalator if Petya counted 20 steps with his feet (before falling) and 30 steps with his sides (after falling)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
In a class consisting of 31 people, each group of 4 students contains at least one student who is friends with the other 3 students. In this class, what is the minimum number of students who are friends with all the other students except themselves?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
28
}
Find the determinant of an $n \times n$ matrix $A$ where each element $a_{i,j}$ is defined as $\gcd(i,j)$. What is the determinant of $A$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\prod_{k=1}^{n} \phi(k)}
Let $J$ , $E$, $R$, and $Y$ be four positive integers chosen independently and uniformly at random from the set of factors of $1428$. What is the probability that $JERRY = 1428$? Express your answer in the form $\frac{a}{b\cdot 2^n}$ where $n$ is a nonnegative integer, $a $and $b$ are odd, and gcd $(a,b) = 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\[ \frac{7}{3 \cdot 2^{12}} \]}
Given that the function $y = (m^2 + 2m - 2)x^{\frac{1}{m-1}}$ is a power function, find the value of $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3}
Given a right circular cone $(P-ABC)$ with lateral edges $(PA)$, $(PB)$, $(PC)$ being pairwise perpendicular, and base edge $AB = \sqrt{2}$, find the surface area of the circumscribed sphere of the right circular cone $(P-ABC)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["3\\pi"]}
In how many ways can a soccer game between two teams end with a total of 12 goals scored, such that the scores of the two teams never differ by more than 2 goals?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{972}
Evaluate the integral \( \int_{0}^{\infty} xe^{-ikx} \, dx \) where \( k \) is a real number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{k^2}}
Evaluate the integral:

$$\frac{\exp\left(-\frac{N}{2}\right)}{2\pi N} \iint \exp\left(-\left( \frac {x^2+2xy+2Ny+2y^2}{2N}\right)\right) \,dx\,dy$$

where $N > 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A traveler token starts on the center tile of a hexagonal grid with 19 tiles. Each turn, a die is rolled to determine the neighboring tile the traveler moves to, with all six directions being equally likely. The game ends on the turn the traveler leaves the board. What is the expected number of turns the game will last?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{55}{9}}
Evaluate the limit of the integral \( \int_0^{\pi/2}\dfrac {dx} {\sqrt{\sin^2(x)+\epsilon \cos^2(x)}} \) as \( \epsilon \to 0^{+} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
In a tournament with 8 players where each player plays every other player exactly once, the winner of a game earns 1 point, the loser earns 0 points, and each player earns 0.5 points if the game is a draw. At the end of the tournament, each player has a different score, and the player who finishes in second place has a score equal to the combined scores of the four players who finish in the bottom four positions. Determine the result of the game between the player who finishes in third place and the player who finishes in seventh place.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
The price of a shirt is increased by 20%, then reduced by 15%, followed by an increase of 10%, and finally, a decrease of 5%. Calculate the final price of the shirt.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.0659}
Determine the conditions on the parameter $m$ such that every nontrivial solution of the differential equation \( y'' + \frac{m}{x}y = 0 \) has an infinite number of positive zeros.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m > 0}
Determine the values of \(a \in \mathbb{R}\) for which the integral \(\int_{0}^{\frac{\pi}{2}} \log(\cos(x)) \cdot (\tan(x))^a \, dx\) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-3, 1)}
Evaluate the improper integral: \[ \int_0^\infty \frac{1}{1+x^2 \sin^2 x} \, dx. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Let $h(x) = 2x - 3$ and $j(x) = \frac{x}{4}$. Calculate the value of \[h(j^{-1}(h^{-1}(h^{-1}(j(h(12)))))).\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{51}{2}}
Given that point M is a moving point on the parabola $y^2=8x$, F is the focus of the parabola, and point A is on the circle $C: (x-3)^2+(y+1)^2=1$, then the minimum value of $|AM|+|MF|$ is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
What is the minimum number of cells that need to be coloured in a 5x5 square grid so that every 1x4 or 4x1 rectangle in the grid has at least one coloured cell? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given three complex numbers $z$, $w$, and $u$ such that $|z|=1$, $|w|=2$, $|u|=3$, and $|z+w-u|=|u+z-w|$, determine the greatest possible value of $|z-u|$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{14}}
Determine whether there exists a natural number $k$ such that for every even integer $n \ge k$, there exist positive integers $a, b, x, y$ with $\gcd(a, b) = 1$ and $a, b > 1$, satisfying $n = a^x + b^y$, where at least one of $x, y$ is greater than $1$. If such a $k$ exists, provide its value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
The very hungry caterpillar lives on the number line. For each non-zero integer $i$, a fruit sits on the point with coordinate $i$. The caterpillar moves back and forth; whenever he reaches a point with food, he eats the food, increasing his weight by one pound, and turns around. The caterpillar moves at a speed of $2^{-w}$ units per day, where $w$ is his weight. If the caterpillar starts off at the origin, weighing zero pounds, and initially moves in the positive $x$ direction, after how many days will he weigh 10 pounds?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9217}
Find the smallest natural number \( n \) such that among any \( n \) irrational numbers, there will always be 3 numbers for which the sum of each pair remains irrational.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given the vector field $F(x,y,z)=(z^2-x,-xy,3z)$ and the surface $S$ of the solid delimited by the equations $z=4-y^2, x=0, x=3$ and $z=0$, with the normal vector exterior, evaluate $\iint_S F\cdot n \,ds$ without using The Divergence Theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
In an $8 \times 8$ chessboard, color the squares using 8 colors such that each square is colored in one color, neighboring squares sharing an edge have different colors, and each row contains all 8 colors. How many different coloring methods are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8! \times 14833^7}
Find all positive integer solutions (x, y) to the equation \(x^4 + 4x^3 + 6x^2 + 4x + 16 = y^2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
Determine the smallest positive integer \( q \) with the following property: for every integer \( m \) with \( 1 \leqslant m \leqslant 1006 \), there exists an integer \( n \) such that

\[ 
\frac{m}{1007} q < n < \frac{m+1}{1008} q 
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2015}
Let $n$ be a positive integer. Let there be $P_{n}$ ways for Pretty Penny to make exactly $n$ dollars out of quarters, dimes, nickels, and pennies. Also, let there be $B_{n}$ ways for Beautiful Bill to make exactly $n$ dollars out of one dollar bills, quarters, dimes, and nickels. As $n$ goes to infinity, the sequence of fractions \frac{P_{n}}{B_{n}}$ approaches a real number $c$. Find $c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
The sequence starts with 12, 13, 15, 17, 111, 113, 117, 119, 123, followed by a number, and then 131. What is the tenth number in the sequence?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{129}
Define the sequence $\left\{a_{n}\right\}$: $a_{1}=1, a_{2}=2, a_{n+2}=a_{n}+a_{n+1}$ for $n \in \mathbf{N}_{+}$. Calculate the value of $\left[\frac{a_{2}}{a_{1}}\right] \cdot\left\{\frac{a_{3}}{a_{2}}\right\} \cdot\left\{\frac{a_{4}}{a_{3}}\right\} \cdot \cdots \ \left\{\frac{a_{99}}{a_{98}}\right\} \cdot\left[\frac{a_{98}}{a_{2}}\right]$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}