inclusionAI/Ring-lite-rl-data huggingface dataset:
Consider an inifinte sequence  $x_1, x_2,\dots$  of positive integers such that, for every integer  $n\geq 1$ :
 
- If  $x_n$  is even,  $x_{n+1}=\dfrac{x_n}{2}$ ;
- If  $x_n$  is odd,  $x_{n+1}=\dfrac{x_n-1}{2}+2^{k-1}$ , where  $2^{k-1}\leq x_n<2^k$ .

Determine the smaller possible value of  $x_1$  for which  $2020$  is in the sequence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1183}
$在\triangle ABC中，内角A，B，C的对边分别为a，b，c，且(\sin B-\sin C)^2=\sin^2A-\sin B\sin C.$
求角A；
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{\pi }{3}$}
It is well-known that the $n^{\text{th}}$ triangular number can be given by the formula $n(n+1)/2$. A Pythagorean triple of $\textit{square numbers}$ is an ordered triple $(a,b,c)$ such that $a^2+b^2=c^2$. Let a Pythagorean triple of $\textit{triangular numbers}$ (a PTTN) be an ordered triple of positive integers $(a,b,c)$ such that $a\leq b<c$ and
\[\dfrac{a(a+1)}2+\dfrac{b(b+1)}2=\dfrac{c(c+1)}2.\]
For instance, $(3,5,6)$ is a PTTN ($6+15=21$). Here we call both $a$ and $b$ $\textit{legs}$ of the PTTN. Find the smallest natural number $n$ such that $n$ is a leg of $\textit{at least}$ six distinct PTTNs.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Integrate $\int \dfrac{\ln(x)}{(1-x^2)^{3/2}}\text{d}x$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{x\ln(x)}{\sqrt{1-x^{2}}} - \arcsin(x) + c}
设对于任意实数x，不等式$\vert x+1\vert+\vert x+a\vert+\vert x+3\vert>5$恒成立，则参数a的取值范围为
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$(-\infty,-2)\cup(6,+\infty)$}
Given that the function f(x) defined on the set of real numbers ℝ satisfies f(x+1) = 1/2 + √(f(x) - f^2(x)), find the maximum value of f(0) + f(2017).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1+\frac{\sqrt{2}}{2}}
Rhona wrote down a list of nine multiples of ten: 10, 20, 30, 40, 50, 60, 70, 80, 90. She then deleted some of these multiples so that the product of the remaining multiples was a square number. What is the least number of multiples that she could have deleted? Provide your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A shepherd uses 15 segments of fencing, each 2 meters long, to form a square or rectangular sheep pen with one side against a wall. What is the maximum area of the sheep pen in square meters?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{112}
Compute $\lim_{z\to 0}z^3\Gamma(z)\Gamma(z-1)\Gamma(z-2)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the number of non-isomorphic groups of order 36 that have a normal subgroup of order 4.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given the cross-section of cone PQ is an equilateral triangle and plane α is parallel to PQ, find the eccentricity of the hyperbola on which the curve obtained by intersecting the cone's lateral surface with plane α lies. Express your answer as a numerical value, including the square root symbol if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2\sqrt{3}}{3}}
Find the smallest positive real number \(\alpha\), such that

\[ \frac{x+y}{2} \geq \alpha \sqrt{x y}+(1-\alpha) \sqrt{\frac{x^{2}+y^{2}}{2}} \]

for all positive real numbers \(x\) and \(y\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Determine the value of the following limit using real analysis methods without complex analysis:

$$\lim_{b \to 0}\int _0^{\infty}\frac{\left(x^2-1\right)\cos \left(bx\right)}{\left(x^2+1\right)^2}dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let  $a$  be the sum of the numbers: $99 \times 0.9$  $999 \times 0.9$  $9999 \times 0.9$  $\vdots$  $999\cdots 9 \times 0.9$ where the final number in the list is  $0.9$  times a number written as a string of  $101$  digits all equal to  $9$ .
Find the sum of the digits in the number  $a$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{891}
Farmer John has a flexible fence of length $L$ and two straight walls that intersect at a corner perpendicular to each other. He knows that if he doesn't use any walls, he can enclose a maximum possible area of $A_0$. When he uses one of the walls or both walls, he gets a maximum area of $A_1$ and $A_2$, respectively. If $n=\frac{A_1}{A_0}+\frac{A_2}{A_1}$, find $\lfloor 1000n \rfloor$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4000}
The edge of cube $A B C D A_{1} B_{1} C_{1} D_{1}$ is equal to 1. Construct a cross-section of the cube by a plane that has the maximum perimeter.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\sqrt{2}}
The distance between two cells of an infinite chessboard is defined as the minimum nuber to moves needed for a king for move from one to the other.One the board are chosen three cells on a pairwise distances equal to  $ 100$ . How many cells are there that are on the distance  $ 50$  from each of the three cells?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
The sequence of integers $a_1, a_2, \dots$ is defined as follows:
- $a_1 = 1$.
- For $n > 1$, $a_{n+1}$ is the smallest integer greater than $a_n$ such that $a_i + a_j \neq 3a_k$ for any $i, j, k$ from $\{1, 2, \dots, n+1\}$, where $i, j, k$ are not necessarily different.

Define $a_{2004}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4507}
Determine the interval where the function y = -x^2 is monotonically increasing. Express your answer as an interval.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 0]}
Compute $\left\lceil\sum_{k=2018}^{\infty}\frac{2019!-2018!}{k!}\right\rceil$. The notation $\left\lceil x\right\rceil$ denotes the least integer $n$ such that $n\geq x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2019}
$$\frac{2}{\sqrt{4-3\sqrt[4]{5}+2\sqrt{5}-\sqrt[4]{125}}}=a+\sqrt[b]{c}$$
Find the value of $a+b+c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
In square \( ABCD \), diagonals \( AC \) and \( BD \) intersect at point \( O \). Point \( K \) is the midpoint of side \( AB \). Points \( M \) and \( N \) are chosen on sides \( AD \) and \( BC \) respectively, such that rays \( OK \), \( OM \), and \( ON \) divide the square into three regions of equal area. In what ratio does point \( M \) divide side \( AD \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5:1}
1. The numbers from 1 to 2150 are written on a board. Every minute, each number undergoes the following operation: if the number is divisible by 100, it is divided by 100; if it is not divisible by 100, 1 is subtracted from it. Find the largest number on the board after 87 minutes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2012}
Given two circles: $x^{2}+y^{2}+2ax+a^{2}-9=0$ and $x^{2}+y^{2}-4by-1+4b^{2}=0$, they have three shared tangents. If $a\in R$, $b\in R$, and $ab\neq 0$, determine the minimum value of $\frac{4}{a^{2}}+\frac{1}{b^{2}}$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Eight red boxes and eight blue boxes are randomly placed in four stacks of four boxes each. The probability that exactly one of the stacks consists of two red boxes and two blue boxes is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{843}
11. Given positive integers $a, b, c, d$ satisfy $a^{2}+2 a=c d+11 c+d+10, b^{2}-2 b=c d-11 c+d-12$, then the sum of all possible values of $d$ is $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
Given the function f(x) = sin(ωx + φ) where ω > 0 and |φ| < π/2, it is known that the smallest positive period is 4π, and f(π/3) = 1. Find the coordinates of a symmetry center of f(x). Express your answer as an ordered pair (x, y), where x is a specific numerical value and y is 0.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\frac{2\pi}{3}, 0)}
Let \( C_1 \) and \( C_2 \) be two concentric circles, with \( C_2 \) inside \( C_1 \). Let \( A \) be a point on \( C_1 \) and \( B \) a point on \( C_2 \) such that the line segment \( AB \) is tangent to \( C_2 \). Let \( C \) be the second point of intersection of the line segment \( AB \) with \( C_1 \), and let \( D \) be the midpoint of \( AB \). A line passing through \( A \) intersects \( C_2 \) at points \( E \) and \( F \) such that the perpendicular bisectors of \( DE \) and \( CF \) intersect at a point \( M \) on the line segment \( AB \). Determine the ratio \( AM / MC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Three speed skaters have a friendly "race" on a skating oval. They all start from the same point and skate in the same direction, but with different speeds that they maintain throughout the race. The slowest skater does $1$ lap per minute, the fastest one does $3.14$ laps per minute, and the middle one does $L$ laps a minute for some $1 < L < 3.14$. The race ends at the moment when all three skaters again come together to the same point on the oval (which may differ from the starting point.) Determine the number of different choices for $L$ such that exactly $117$ passings occur before the end of the race.
Note: A passing is defined as when one skater passes another one. The beginning and the end of the race when all three skaters are together are not counted as passings.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
"渐升数"是指每个数字比其左边的数字大的正整数, 如 34689 , 已知有 $C_9^5=126$ 个五位
"渐升数", 若把这些数按从小到大的顺序排列, 则第 100 个数是 $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24789}
There are originally 5 different books placed side by side on a bookshelf. Now, if we want to insert 3 different books, one at a time, into the gaps formed by the existing books, how many different insertion methods are there in total? Express your answer as a numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{336}
Evaluate the integral $$\iiint_K e^z \, dx\, dy\, dz$$ where $K$ is the solid ball defined by $$K = \{x^2 + y^2 + z^2 \leq 1\}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{4\pi}{e}\)}
Given the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ (where $a > 0, b > 0$), a tangent line $FM$ to the circle $x^{2} + y^{2} = a^{2}$ is drawn from the right focus $F$ of the hyperbola, intersecting the y-axis at point $P$. If $M$ is the midpoint of segment $FP$, find the eccentricity of the hyperbola.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Let $m \ge 3$ be an integer and let $S = \{3,4,5,\ldots,m\}$. Find the smallest value of $m$ such that for every partition of $S$ into two subsets, at least one of the subsets contains integers $a$, $b$, and $c$ (not necessarily distinct) such that $ab = c$. <b>Note</b>: a partition of $S$ is a pair of sets $A$, $B$ such that $A \cap B = \emptyset$, $A \cup B = S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$243$}
Siva has the following expression, which is missing operations:
$$\frac{1}{2} \_ \frac{1}{4} \_ \frac{1}{8} \_ \frac{1}{16} \_ \frac{1}{32}.$$ 

For each blank, he flips a fair coin: 
- If it comes up heads, he fills it with a plus.
- If it comes up tails, he fills it with a minus.

Afterwards, he computes the value of the expression. He then repeats the entire process with a new set of coin flips and operations. 

If the probability that the positive difference between his computed values is greater than $\frac{1}{2}$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$, $b$, then find $a + b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{39}
Compute the decimal expansion of \(\sqrt{\pi}\). Your score will be \(\min (23, k)\), where \(k\) is the number of consecutive correct digits immediately following the decimal point in your answer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.77245385090551602729816}
A fly and $\mathrm{k}$ spiders move on a $2019 \times 2019$ grid. On their turn, the fly can move 1 square and the $k$ spiders can each move 1 square. What is the minimal $k$ for which the spiders are sure to catch the fly.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $t$ represent all real values. Consider the curve formed by the set of points $(x, y)$ defined by the equations:
\[
x = t^3 - 3t + 1, 
\]
\[
y = t^4 - 4t^2 + 4.
\]
Determine the ordered pair $(x,y)$ where the curve crosses itself.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1,1)}
Given Tom paid $180, Dorothy paid $200, Sammy paid $240, and Alice paid $280, and they agreed to split the costs evenly, calculate the amount that Tom gave to Sammy minus the amount that Dorothy gave to Alice.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Two positive integers $p,q \in \mathbf{Z}^{+}$ are given. There is a blackboard with $n$ positive integers written on it. A operation is to choose two same number $a,a$ written on the blackboard, and replace them with $a+p,a+q$. Determine the smallest $n$ so that such operation can go on infinitely.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{p+q}{\gcd(p,q)}}
What is the largest $2$-digit prime factor of the integer $n = {300 \choose 150}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{97}
Given a semicircle O whose diameter lies along the base AB of a trapezoid ABCD, and BC, CD, DA are tangent to the semicircle, find the length of AB. The lengths of BC and DA are 2 and 3, respectively. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Let $M=\{1,2,3,\ldots,1995\}$, and let $A$ be a subset of $M$ that satisfies the condition: if $x \in A$, then $15x \notin A$. The maximum number of elements in $A$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1870}
Zebadiah has 3 red shirts, 3 blue shirts, and 3 green shirts in a drawer. Without looking, he randomly pulls shirts from his drawer one at a time. What is the minimum number of shirts that Zebadiah must pull out to guarantee that he has either 3 shirts of the same color or 3 shirts of different colors? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Let $\triangle ABC$ be a triangle with $AB = 13$, $BC = 14$, and $CA = 15$. The altitude from $A$ intersects $BC$ at $D$.

Let $\omega_1$ and $\omega_2$ be the incircles of $\triangle ABD$ and $\triangle ACD$, respectively. The common external tangent of $\omega_1$ and $\omega_2$ (other than $BC$) intersects $AD$ at $E$. Compute the length of $AE$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
It is known that the quadratic equations $a x^{2} + b x + c = 0$ and $b x^{2} + c x + a = 0$ (where $a$, $b$, and $c$ are non-zero numbers) have a common root. Find this root.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
## Task 1

The members of a youth brigade helped to prepare apartments for elderly citizens. The value amounted to 1200 Marks. They helped with cable installation. This value amounted to 2100 Marks. In the operation, they saved materials. This value amounted to 1500 Marks.

How many Marks did the youth brigade bill for?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4800}
Given a function $f(x) = (m^2 - m - 1)x^{m^2 - 2m - 1}$ which is a power function and is increasing on the interval $(0, \infty)$, find the value of the real number $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
In the dihedral angle $\alpha - E F - \beta $, $AE \subset \alpha, BF \subset \beta$, and $AE \perp EF, BF \perp EF$. Given $EF = 1$, $AE = 2$, and $AB = \sqrt{2}$, find the maximum volume of the tetrahedron $ABEF$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
An ant walks inside a 18 cm by 150 cm rectangle. The ant's path follows straight lines which always make angles of 45° to the sides of the rectangle. The ant starts from a point X on one of the shorter sides. The first time the ant reaches the opposite side, it arrives at the midpoint. Calculate the distance, in centimeters, from X to the nearest corner of the rectangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
An insect lives on the surface of a regular tetrahedron with edges of length 1. It wishes to travel on the surface of the tetrahedron from the midpoint of one edge to the midpoint of the opposite edge. What is the length of the shortest such trip? (Note: Two edges of a tetrahedron are opposite if they have no common endpoint.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$1$}
3. Petrov lists the odd numbers: $1,3,5, \ldots, 2013$, while Vasechkin lists the even numbers: $2,4, \ldots, 2012$. Each of them calculated the sum of all digits of all their numbers and told the excellent student Masha. Masha subtracted Vasechkin's result from Petrov's result. What did she get?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1007}
Consider two parallel lines $a$ and $b$. The circles $C, C_1$ are tangent to each other and to the line $a$. The circles $C, C_2$ are tangent to each other and to the line $b$. The circles $C_1, C_2$ are tangent to each other, with radii $R_1 = 9$ and $R_2 = 16$. What is the radius $R$ of the circle $C$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
设 $\left\{a_{n}\right\}$ 为等差数列， $\left\{b_{n}\right\}$ 为等比数列，且 $b_{1}=a_{1}{ }^{2}, b_{2}=a_{2}{ }^{2}, b_{3}=a_{3}{ }^{2}\left(a_{1}<a_{2}\right)$ ，又 $\lim \limits_{n \rightarrow+\infty}\left(b_{1}+b_{2}+\cdots+b_{n}\right)=\sqrt{2}+1$. 试求 $\left\{a_{n}\right\}$ 的2倍首项与公差的和.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
6.2. On the sides $AB$ and $AC$ of triangle $ABC$, points $E$ and $F$ are taken, respectively. Segments $BF$ and $CE$ intersect at point $D$. Find the area of triangle $AEF$, if the areas of triangles $BCD$, $BDE$, and $CDF$ are $1$, $\frac{1}{2}$, and $\frac{1}{6}$, respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7}{44}}
In a 6-deck game of cards (each deck contains 52 cards), what is the probability of drawing a spade and specifically the King of Diamonds when drawing two cards consecutively without replacement?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3/622}
There has been successive increases of 15% and then 10% in the price of gas from the previous month. By what percentage should a driver reduce gas consumption so that the expenditure does not change?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20.95\%}
A line intersects a circle with radius \( R \) at points \( A \) and \( B \) such that the measure of arc \( \overset{\frown}{AB} = 45^\circ \). Additionally, a line perpendicular to the diameter \( AM \) of the circle and passing through its center intersects the first line at point \( D \). The line passing through point \( B \) perpendicular to the diameter \( AM \) intersects \( AM \) at point \( C \). Find the area of the trapezoid \( OCBD \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{R^2(3 + \sqrt{2})}{4}}
Let $ABCD$ be an isosceles trapezoid with bases $AB=92$ and $CD=19$. Suppose $AD=BC=x$ and a circle with center on $\overline{AB}$ is tangent to segments $\overline{AD}$ and $\overline{BC}$. If $m$ is the smallest possible value of $x$, then $m^2$=
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$1679$}
A certain number of men are equal to as many women as are equal to 8 boys. All of them earn Rs. 105 only. Men's wages are Rs. 7. How many men are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
A straight line is drawn on an 8 x 8 chessboard. What is the largest possible number of the unit squares with interior points on the line?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Let points $A = (0,0)$, $B = (2,4)$, $C = (6,6)$, and $D = (8,0)$. Quadrilateral $ABCD$ is cut into two pieces by a line passing through $A$ and intersecting $\overline{CD}$ such that the area above the line is twice the area below the line. This line intersects $\overline{CD}$ at a point $\left(\frac{p}{q}, \frac{r}{s}\right)$, where these fractions are in lowest terms. Calculate $p + q + r + s$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
A local college is offering German lessons and currently has 8 students enrolled. For the first month, they run an advertising campaign that increases enrollment by 30%, with a fourth of these new students dropping out within the first week. Meanwhile, a nearby school hosts a cultural exchange festival and 5 international students decide to join the class, but 10% of them leave within two weeks. In the second month, the class holds an open house event which increases enrollment by 25% of the students already enrolled, but 3 of these new students dropped out due to scheduling conflicts.

During the third month, the class receives a boost in enrollment when a high school's language club decides to join the course. This adds an enrollment boom that's equal to 50% of the total current students, but 20% of them eventually quit due to the difficulty level. By the end of the third month, two-thirds of all remaining students graduate.

The college plans to run one last advertising campaign in the fourth month, with an expected 20% increase in enrollment. However, they also predict within this month that 30% of the new enrollees will not be able to keep up with the course material and drop out, while 10% of all the remaining students will ultimately graduate. How many students are expected to still be enrolled in the German lessons at the end of the fourth month?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Three sectors of a circle are removed from a regular hexagon to form a shaded shape. Each sector has a perimeter of 18 mm. Calculate the perimeter, in millimeters, of the shaded shape formed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
A boy runs 4.8 miles in 33 minutes. What is his speed in kilometers per hour?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14.05}
A circle has its center at $(2,0)$ with a radius of 2, and another circle has its center at $(5,0)$ with a radius of 1. A line is tangent to both circles in the first quadrant. The $y$-intercept of this line is closest to:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2 \sqrt{2}$}