inclusionAI/Ring-lite-rl-data huggingface dataset:
Find the infimum of the set of distances \(\left\{\int_0^1 |A'(t)|\,dt + \int_0^1 |B'(t)|\,dt: (A,B)\in S\right\}\), where \(S\) consists of pairs of smooth paths \(A,B: [0,1]\rightarrow\mathbb{R}^2\) satisfying \(A(0) = (0,0)\), \(B(0) = (1,0)\), and \(B(1) - A(1) = (1,0)\). This infimum represents the shortest total distance that two points in the plane, initially separated by a distance of 1 in the horizontal direction, must travel to become vertically separated by the same distance.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Six friends earn $25, $30, $35, $45, $50, and $60. Calculate the amount the friend who earned $60 needs to distribute to the others when the total earnings are equally shared among them.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19.17}
Determine the center of the Lie algebra $\mathfrak{gl}_n(\mathbb{F})$ when $\mathbb{F}$ is a field with positive characteristic $p$ and $n$ is an integral multiple of $p$. Provide your answer in terms of $n$ and $p$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{F} I_n}
Determine the value of $t$ such that the sum of the probabilities $P(T > -t)$, $P(T > 0)$, $P(T > t)$, and $P(T > 2t)$ for a random variable $T$ from a Student-$t$ distribution with 7 degrees of freedom equals 1.75.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.356}
In a hockey match, the away team was leading at half-time with a total of six goals scored in the first half. In the second half, the home team scored three goals and won the game. How many goals did the home team score altogether?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
如图， $A B$ 是单位圆的直径，在 $A B$ 上任取一点 $D$ ，作 $D C \perp A B$, 交圆周于 $C$ 。若点 $D$ 的坐标为 $(x, 0)$,则当 $x \in$ $\qquad$时，线段 $A D 、 B D 、 C D$ 可以构成锐角三角形。请提供区间的两个端点之和。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Convert $1729_{10}$ to base 6.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12001_6}
A target consists of five zones: the center circle (bullseye) and four colored rings. The width of each ring is equal to the radius of the bullseye. It is known that the score for hitting each zone is inversely proportional to the probability of hitting that zone, and hitting the bullseye is worth 315 points. How many points is hitting the blue (second-to-last) zone worth?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
What is the greatest number of integers that can be selected from a set  of  $2015$  consecutive numbers so that no sum of any two selected numbers is divisible by their difference?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{672}
Xiao Ming departs from his home to his grandmother's house on a bicycle at a speed of 12 km/h. After 2.5 hours, his father realizes that Xiao Ming forgot to bring his homework and starts chasing him on a motorcycle at a speed of 36 km/h. Half an hour after Xiao Ming reaches his grandmother's house, his father arrives. What is the distance between Xiao Ming's home and his grandmother's house in kilometers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36 \ \text{km}}
Find the largest positive integer  $n$  with the following property: there are rectangles  $A_1, ... , A_n$  and  $B_1,... , B_n,$  on the plane , each with sides parallel to the axis of the coordinate system, such that the rectangles  $A_i$  and  $B_i$  are disjoint for all  $i \in \{1,..., n\}$ , but the rectangles  $A_i$  and  $B_j$  have a common point for all  $i, j \in \{1,..., n\}$ ,  $i \ne j$ .

*Note: By points belonging to a rectangle we mean all points lying either in its interior, or on any of its sides, including its vertices*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
The sum of Jim's weight and Bob's weight is 180 pounds. If you subtract Jim's weight from Bob's weight, you get half of Bob's weight. How many pounds does Bob weigh?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Given that the positive real numbers \(a, b, c\) satisfy the equation \(a^{2}+4 b^{2}+9 c^{2}=4 b+12 c-2\), find the minimum value of \(\frac{1}{a}+\frac{2}{b}+\frac{3}{c}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given the function
$$
f(x) = \left|8x^3 - 12x - a\right| + a
$$

The maximum value of this function on the interval \([0, 1]\) is 0. Find the maximum value of the real number \(a\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2\sqrt{2}}
Let \( A \subseteq \{0, 1, 2, \cdots, 29\} \) such that for any integers \( k \) and any numbers \( a \) and \( b \) (possibly \( a = b \)), the expression \( a + b + 30k \) is not equal to the product of two consecutive integers. Determine the maximum possible number of elements in \( A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Determine the maximum possible value of $F(x)$ for functions $F(x)$ satisfying $F(x)=f(x)^2+f'(x)^2$ such that $|f(x)| \le 2$ and $f(x)$ is thrice differentiable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let \( n = \overline{abc} \) be a three-digit number, where \( a, b, \) and \( c \) are the digits of the number. If \( a, b, \) and \( c \) can form an isosceles triangle (including equilateral triangles), how many such three-digit numbers \( n \) are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{165}
Let $k$ be the product of every third positive integer from $2$ to $2006$, that is, \[ k = 2 \cdot 5 \cdot 8 \cdot 11 \cdots 2006. \] Find the number of zeros at the right end of the decimal representation for $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{168}
The function $f: \mathbb{Z}^{2} \rightarrow \mathbb{Z}$ satisfies - $f(x, 0)=f(0, y)=0$, and - $f(x, y)=f(x-1, y)+f(x, y-1)+x+y$ for all nonnegative integers $x$ and $y$. Find $f(6,12)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{77500}
Call a $7$-digit telephone number $d_1d_2d_3-d_4d_5d_6d_7$ memorable if the prefix sequence $d_1d_2d_3$ is exactly the same as either of the sequences $d_4d_5d_6$ or $d_5d_6d_7$ (possibly both). Assuming that each $d_i$ can be any of the ten decimal digits $0, 1, 2, \ldots, 9$, the number of different memorable telephone numbers is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19990}
Given a differentiable function $f(x)$ defined on the interval $(0,+\infty)$, with its derivative denoted by $f'(x)$, and satisfying $xf'(x) + 2f(x) > 0$, find the solution set of the inequality $\frac{(x+2016)f(x+2016)}{5} < \frac{5f(5)}{x+2016}$. Express your answer in interval notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-2016, -2011)}
Compute the telescoping sum $\sum_{n=2}^{\infty}\frac 12 (\frac{1}{4.6…(3n-2)} - \frac{1}{4.6…(3n+1)})$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/48}
Given a trapezoid \(ABCD\) with bases \(AD = a\) and \(BC = b\). Points \(M\) and \(N\) lie on sides \(AB\) and \(CD\) respectively, with the segment \(MN\) parallel to the bases of the trapezoid. Diagonal \(AC\) intersects this segment at point \(O\). Find \(MN\), given that the areas of triangles \(AMO\) and \(CNO\) are equal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{ab}}
A regular pentagon with area \(\sqrt{5}+1\) is printed on paper and cut out. The five vertices of the pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon? The original answer is in the form of \(n - \sqrt{m}\). Please find the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
A thin piece of wood of uniform density in the shape of an equilateral triangle with side length $3$ inches weighs $12$ ounces. A second piece of the same type of wood, with the same thickness, also in the shape of an equilateral triangle, has a side length of $5$ inches. Find the weight, in ounces, of the second piece that is closest to an integer value. If the weight is w ounces, please provide the value of 10w.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{333}
A square flag features a green cross of uniform width, and a yellow square in the center, against a white background. The cross is symmetric with respect to each of the diagonals of the square. Suppose the entire cross (including the green arms and the yellow center) occupies 49% of the area of the flag. If the yellow center itself takes up 4% of the area of the flag, what percent of the area of the flag is green?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45\%}
If some percentage of (x - y) = 20% of (x + y), then y is 42.857142857142854% of x. What is the percentage of (x - y)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50\%}
We spill the content of a box of paprika-flavored and a box of bear's garlic-flavored, eight-slice Maci cheese onto a table, then put the 16 small cheese slices (with their labels facing up) back into the boxes. In how many different ways can we do this if arrangements that can be transformed into each other by rotation are not considered different, but the two boxes are distinguished from each other?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{234}
There were some roses and 3 orchids in the vase. Jessica cut some more roses and orchids from her flower garden. There are now 12 roses and 2 orchids in the vase. There are 10 more roses than orchids in the vase now. How many roses were there in the vase initially?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In $\triangle ABC$, $\sin^{2} A - \sin^{2} C = (\sin A - \sin B) \sin B$. Find the measure of angle $C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{3}}
Let  $ S \equal{} \{1,2,\ldots,2008\}$ . For any nonempty subset  $ A\in S$ , define  $ m(A)$  to be the median of  $ A$  (when  $ A$  has an even number of elements,  $ m(A)$  is the average of the middle two elements). Determine the average of  $ m(A)$ , when  $ A$  is taken over all nonempty subsets of  $ S$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2009}{2}}
Let a,b,c, and d be real numbers such that \[a^2 + 3b^2 + \frac{c^2+3d^2}{2} = a + b + c+d-1.\] Find $1000a + 100b + 10c + d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{527}
Let a constant \( a \in \mathbf{R} \) and the function
\[ f(x) = (a - x)|x| \]

have an inverse function \( f^{-1}(x) \). If the inequality
\[ f^{-1}\left(x^{2} + m\right) < f(x) \]

holds for all \( x \in [-2,2] \), find the range of values for the real number \( m \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m \in (12, +\infty)}
Line segments drawn from the vertex opposite the hypotenuse of a right triangle to the points trisecting the hypotenuse have lengths $\sin x$ and $\cos x$, where $x$ is a real number such that $0<x<\frac{\pi}{2}$. The length of the hypotenuse is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3\sqrt{5}}{5}}
If $\sqrt[3]{0.3}\approx 0.6694$ and $\sqrt[3]{3}\approx 1.442$, then $\sqrt[3]{300}\approx$____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6.694}
A fat coin is one which, when tossed, has a $2 / 5$ probability of being heads, $2 / 5$ of being tails, and $1 / 5$ of landing on its edge. Mr. Fat starts at 0 on the real line. Every minute, he tosses a fat coin. If it's heads, he moves left, decreasing his coordinate by 1; if it's tails, he moves right, increasing his coordinate by 1. If the coin lands on its edge, he moves back to 0. If Mr. Fat does this ad infinitum, what fraction of his time will he spend at 0?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\[
\frac{1}{3}
\]}
Congcong performs a math magic trick by writing the numbers $1, 2, 3, 4, 5, 6, 7$ on the blackboard and lets others select 5 of these numbers. The product of these 5 numbers is then calculated and told to Congcong, who guesses the chosen numbers. If when it is Benben's turn to select, Congcong cannot even determine whether the sum of the 5 chosen numbers is odd or even, what is the product of the 5 numbers chosen by Benben?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{420}
Jesse and Mia are competing in a week long race. They have one week to run 30 miles. On the first three days Jesse averages (2/3) of a mile. On day four she runs a certain distance. Mia averages 3 miles a day over the first 4 days. The average of their average that they have to run over the final three days is 6 miles. How many miles did Jesse run on day four?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Let $S = \{1, 2, \ldots, 100\}$. Consider a partition of $S$ into $S_1, S_2, \ldots, S_n$ for some $n$, i.e., $S_i$ are nonempty, pairwise disjoint sets, and $S = \bigcup_{i=1}^n S_i$. Let $a_i$ be the average of the elements of the set $S_i$. Define the score of this partition by:
\[
\frac{a_1 + a_2 + \ldots + a_n}{n}.
\]

Among all $n$ and partitions of $S$, determine the minimum possible score.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
In a village of 2011 dwarfs, each dwarf is either wearing a red hat or a blue hat. Dwarfs with red hats always tell the truth, while dwarfs with blue hats can lie. Additionally, they can change the color of their hats at any time. One day, each dwarf encountered every other dwarf exactly once, and each one claimed that the other was wearing a blue hat. What is the minimum number of times the dwarfs changed the color of their hats that day?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2009}
On the first day, Johnny spent 3 hours working on a job that paid $7 per hour, 2 hours working on a job that paid $10 an hour, and 4 hours working on a job that paid $12 an hour. On the second day, his pay rate for the first job increased by $2, for the second job it decreased by $1, and the pay rate for the third job stayed the same. For the next three days (days 3, 4, and 5), his pay rates changed as follows: on day 3, the pay rates for the first two jobs returned to their initial rates, and the third job pay increased by $3; on day 4, the first job pay decreased by $3, the second job pay increased by $2, and the third job pay remained the same as day 3; and on day 5, the pay rates for all jobs returned to their initial rates. 

Johnny also has work-related expenses, which include a daily $5 lunch and a weekly $20 transportation cost. Additionally, he must pay 15% taxes on his total earnings for the week. Calculate Johnny's net income for the week considering his pay rates, taxes, and work-related expenses.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{352.80}
A convex broken line of length \( d \) rotates around a straight line connecting its ends. What can the area of the surface of rotation obtained in this way be?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi d^{2}}{2}}
There are three solutions with different percentages of alcohol. If you mix them in a ratio of $1:2:3$, you get a $20\%$ solution. If you mix them in a ratio of $5:4:3$, you will get a solution with $50\%$ alcohol content. What percentage of alcohol will the solution contain if equal amounts of the original solutions are mixed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Jeanette is practicing her juggling. Each week she can juggle 2 more objects than the week before. She practices 3 times a week, and with each practice session, she can juggle 1 extra object than the previous session that week. If she starts out juggling 3 objects and practices for 5 weeks, how many objects can she juggle by the end of the 5th week?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
In the following diagram, two lines that meet at a point \( A \) are tangent to a circle at points \( B \) and \( C \). The line parallel to \( AC \) passing through \( B \) meets the circle again at \( D \). Join the segments \( CD \) and \( AD \). Suppose \( AB = 49 \) and \( CD = 28 \). Determine the length of \( AD \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{63}
Evaluate the limit of the double integral as \( n \to \infty \):

\[ \lim_{n \to \infty} \int_{x=-\infty}^{x=\infty} \int_{y=x-\frac{1}{n}}^{y=x+\frac{1}{n}} dy\ dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Find all real values of $y$ that satisfy $\frac{1}{y(y+2)} - \frac{1}{(y+2)(y+4)} < \frac{1}{4}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, -4) \cup (-2, 0) \cup (2, \infty)}
求最大的正整数 E，使得存在一个有 60 个顶点、E 条边的简单图 G，并且可以将 G 的边红蓝二染色，满足没有长度是 3 或 5 的同色圈.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1350}
Evaluate the limit \( I = \lim_{x \to 0^{+}} \frac{\int_{0}^{x^2} t \sin{(\sqrt{x^2-t^2})} \, dt}{x^3} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the sum of the series \( \sum_{k=1}^{n} (-1)^{k+1} \frac{1}{k} \binom{n}{k} \) for \( n \ge 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{H_n}
Let \(p,\) \(q,\) and \(r\) be constants, and suppose that the inequality \[\frac{(x-p)(x-q)}{x-r} \le 0\] is true if and only if either \(x < -6\) or \(|x-30| \le 2.\) Given that \(p < q,\) find the value of \(p + 2q + 3r.\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{74}
2. Given the line $L: x+y-9=0$ and the circle $M: 2 x^{2}+2 y^{2}-8 x-8 y-1=0$, point $A$ is on line $L$, $B, C$ are two points on circle $M$, in $\triangle A B C$, $\angle B A C=45^{\circ}, A B$ passes through the center of circle $M$, then the range of the x-coordinate of point $A$ is $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\leqslantx\leqslant6}
there is a church tower 150 feet tall and another catholic tower at a distance of 350 feet from it which is 200 feet tall . there is one each bird sitting on top of both the towers . they fly at a constant speed and time to reach a grain in b / w the towers at the same time . at what distance from the church is the grain ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
Given $\angle A = 55^\circ$, the complement of $\angle A$ is ____ degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
There are 4 puppies and 3 dogs at a camp. Each dog eats a certain amount of food compared to a puppy, but each puppy eats three times as often as a dog. If a dog eats 4 pounds of food three times a day, the total amount of food the dogs and puppies eat in a day is 108 pounds. What is the ratio of the amount of food a dog eats to the amount a puppy eats?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2:1}
Find the dimension of the quotient ring $\mathbb{C}[x,y]/\left\langle x^3-y^2, x^2y-y^3\right\rangle$ as a vector space over $\mathbb{C}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
The product of four different positive integers is equal to $11^4$. What is the sum of these four integers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}
Point \( K \) lies on side \( AB \) of triangle \( ABC \) with a \(120^\circ\) angle at vertex \( C \). Circles are inscribed in triangles \( AKC \) and \( BKC \) with centers \( O \) and \( Q \), respectively. Find the radius of the circumcircle of triangle \( OQC \), given that \( OK = 6 \) and \( KQ = 7 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\frac{85}{3}}}
Given that the random variable X follows a normal distribution X ~ N(1, 4), and P(X ≤ 0) = 0.1, calculate the probability P(0 < X < 2). Express your answer as a decimal value between 0 and 1, rounded to 1 decimal place if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.8}
A reduction in the price of petrol by 15% enables a motorist to buy 7 gallons more for $300. Find the original price of petrol?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$7.56}
Given that \( A \) and \( B \) are two points on the surface of a sphere with a radius of 5, and \( AB = 8 \). Planes \( O_1AB \) and \( O_2AB \) are perpendicular to each other and pass through \( AB \). The intersections of these planes with the sphere create cross-sections \(\odot O_1\) and \(\odot O_2\). Let the areas of \(\odot O_1\) and \(\odot O_2\) be denoted as \( S_1 \) and \( S_2 \) respectively. Solve for \( S_1 + S_2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41 \pi}
Evaluate the integral:
$$\int_{0}^{\frac{\pi}{2}} x\ln(\tan(x))dx $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Consider the set $\Omega$ consisting of all bijections $\phi$ from $\mathbb{N} \cup \{0\}$ to $\mathbb{Q}$. Each bijection $\phi$ can be visualized as a "staircase" in which, for any $x \in \mathbb{R}^+$, $x$ maps to $\phi(\lfloor x \rfloor)$. The largest step in this staircase is given by $\sup_{n \geq 0} |\phi(n+1)-\phi(n)|$. Determine the smallest possible value for this largest step, formally expressed as:
\[ \inf_{\phi \in \Omega} \sup_{n \geq 0} |\phi(n+1)-\phi(n)| \ \ ? \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
On a table, there are 100 piles of pebbles, with 1, 2, ..., up to 100 pebbles in each pile respectively. In one step, you can reduce any number of selected piles, provided that you remove the same number of pebbles from each selected pile.

What is the minimum number of steps needed to remove all the pebbles from the table?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Given proposition p: The solution set of the equation $4x^2-2ax+2a+5=0$ has at most two subsets, and proposition q: $1-m \leq x \leq 1+m$, $m>0$. If $\neg p$ is a necessary but not sufficient condition for $\neg q$, find the range of the real number $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[9, +\infty)}
Given $|\vec{a}|=1$, $|\vec{b}|=6$, and $\vec{a}\cdot(\vec{b}-\vec{a})=2$, calculate the angle between $\vec{a}$ and $\vec{b}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{3}}
对任意正实数 $a$ ，记函数 $f(x)=|\lg x|$ 在 $[a,+\infty)$ 上的最小值为 $m_{a}$, 函数 $g(x)=\sin \frac{\pi x}{2}$ 在 $[0, a]$ 上的最大值为 $M_{a}$. 若 $M_{a}-m_{a}=\frac{1}{2}$, 求 $a$ 的所有可能值。已知原始答案形式为 \frac{m}{n} 或 \sqrt{m}，请给出 m + n 的最大值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}