inclusionAI/Ring-lite-rl-data huggingface dataset:
Let $a$ and $b$ be nonnegative real numbers such that
\[\sin (ax + b) = \sin 29x\]for all integers $x.$  Find the smallest possible value of $a.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10 \pi - 29}
Given the equation \(\frac{\left(\frac{a}{c}+\frac{a}{b}+1\right)}{\left(\frac{b}{a}+\frac{b}{c}+1\right)}=11\), where \(a, b\), and \(c\) are positive integers, find the number of different ordered triples \((a, b, c)\) such that \(a+2b+c \leq 40\). Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Vasya wrote a note on a piece of paper, folded it in quarters, wrote "MAME" on top, then unfolded the note, added something else, folded it again along the crease lines randomly (not necessarily as before), and left it on the table with a random side facing up. Find the probability that the "MAME" inscription is still on top.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{8}}
An equilateral triangle is placed on top of a square with each side of the square equal to one side of the triangle, forming a pentagon. What percent of the area of the pentagon is the area of the equilateral triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4\sqrt{3} - 3}{13} \times 100\%}
S is the set of all $(a, b, c, d, e, f)$ where $a$, $b$, $c$, $d$, $e$, $f$ are integers such that $a^2 + b^2 + c^2 + d^2 + e^2 = f^2$. Find the largest $k$ which divides $abcdef$ for all members of $S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
In a school there are  $1200$  students. Each student is part of exactly  $k$  clubs. For any  $23$  students, they are part of a common club. Finally, there is no club to which all students belong. Find the smallest possible value of  $k$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
In a certain company, there are full-time and part-time employees. Among the full-time employees, 60% are men and 40% are women. Among the part-time employees, 45% are men and 55% are women. During the annual company picnic, 20% of the full-time men, 30% of the full-time women, 25% of the part-time men, and 35% of the part-time women attended. If full-time employees make up 70% of the total workforce, what percentage of all employees attended the annual company picnic?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26\%}
Consider a circle of radius \( R \) centered at the origin on the Cartesian plane. Specify at least one value of \( R \) for which there are exactly 32 integer points (a point is called an integer point if both its x-coordinate and y-coordinate are integers) lying on this circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{1105}}
Each hotel room can accommodate no more than 3 people. The hotel manager knows that a group of 100 football fans, who support three different teams, will soon arrive. A room can only house either men or women; and fans of different teams cannot be housed together. How many rooms need to be booked to accommodate all the fans?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
袋内有 8 个白球和 2 个红球，每次从中随机取出一个球，然后放回一个白球，则第 4 次恰好取完所有红球的概率为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5217}
从正整数列1,2,3,4，···中划去3及4的倍数，但其中凡是5的倍数均保留（例如15,20,60,120均不划去），划完后，剩下的数依原有次序组成一个数列：$a _ { 1 } = 1,a _ { 2 } =2,a _ { 3 } = 5 ,a _ { 4 } = 7, \cdots$.试求$a _ { 2 0 0 3 } .$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3338}
Given the mapping function f: A → B defined as f(n) = 2n^2, where A and B are subsets of the natural numbers, find the original element n in set A that maps to 20 in set B. Express your answer as a single natural number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
There are \( N \) lockers, labeled from 1 to \( N \), placed in clockwise order around a circular hallway. Initially, all lockers are open. Ansoon starts at the first locker and always moves clockwise. When she is at locker \( n \) and there are more than \( n \) open lockers, she keeps locker \( n \) open and closes the next \( n \) open lockers, then repeats the process with the next open locker. If she is at locker \( n \) and there are at most \( n \) lockers still open, she keeps locker \( n \) open and closes all other lockers. She continues this process until only one locker is left open. What is the smallest integer \( N > 2021 \) such that the last open locker is locker 1?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2046}
## Task 2 - 191242

Let $M$ be the set of all square areas $Q$ that lie in a given plane $\epsilon$, have a given point $Z$ of the plane $\epsilon$ as their center, and have a given segment length $a$ as their side length.

For arbitrary square areas $Q, Q^{\prime}$ from this set $M$, let $u\left(Q \cup Q^{\prime}\right)$ denote the perimeter of the polygonal area that results from the intersection of the square areas $Q$ and $Q^{\prime}$.

Investigate whether there are square areas $Q, Q^{\prime}$ in $M$ with the smallest possible $u\left(Q \cap Q^{\prime}\right)$.

If this is the case, determine (in dependence on $a$) this smallest possible value of $u\left(Q \cap Q^{\prime}\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8a(\sqrt{2}-1)}
Problem 40. Find the best positive real constant $k$ such that the following inequality holds for all positive real numbers $a, b$ and $c$
$$\frac{(a+b)(b+c)(c+a)}{a b c}+\frac{k(a b+b c+c a)}{a^{2}+b^{2}+c^{2}} \geq 8+k$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4 \sqrt{2}}
Xiaoqing is preparing a dish called "Xiang Qing Fried Eggs," which requires 7 steps with the following times:
- Wash scallions, chop scallions: 1 minute
- Beat eggs: half a minute
- Stir egg mixture and scallions: 1 minute
- Wash pan: half a minute
- Heat pan: half a minute
- Heat oil: half a minute
- Cook dish: 2 minutes

The minimum time to complete this dish is $\qquad$ minutes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5 \text{ minutes}}
Find the limit of the sequence: $$\lim_{n\to \infty} \sqrt[n]{\prod_{k=3}^{n+2}\left(1-\frac{3}{k}\right)^{k}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-3}}
Rachel and Steven play games of chess. If either wins two consecutive games, they are declared the champion. The probability that Rachel will win any given game is 0.6, the probability that Steven will win any given game is 0.3, and the probability that any given game is drawn is 0.1. Find the value of \(1000P\), where \(P\) is the probability that neither is the champion after at most three games.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{343}
Three different numbers are chosen such that when each of the numbers is added to the average of the remaining two, the numbers 65, 69, and 76 result. Find the average of the three original numbers. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
Given the circle \( x^{2} + y^{2} = k^{2} \) at least covers one maximum point and one minimum point of the function \( f(x) = \sqrt{3} \sin \frac{\pi x}{k} \), find the range of the real number \( k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, -2] \cup [2, +\infty)}
Find the numerator of \(\frac{1010\overbrace{11 \ldots 11}^{2011 \text{ ones}}0101}{1100\underbrace{11 \ldots 11}_{2011\text{ ones}}0011}\) when reduced.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9091}
Let  $a_0 = 1$  and define the sequence  $\{a_n\}$  by \[a_{n+1} = \frac{\sqrt{3}a_n - 1}{a_n + \sqrt{3}}.\] If  $a_{2017}$  can be expressed in the form  $a+b\sqrt{c}$  in simplest radical form, compute  $a+b+c$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given a square pyramid \( S-ABCD \) with a height of \( \frac{\sqrt{2}}{4} \), where the base is a square with side length 1 and points \( S, A, B, C, D \) all lie on the surface of a sphere with radius 1. Find the maximum length of the slant edge \( SA \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{6 + 2 \sqrt{7}}}{2}}
求最大的整数 k\leq 2023，满足如下条件：当 A 将 \{1,2,\cdots,2023\} 中的任意 k 个数染红时，B 可将余下若干个数染蓝，使得红数之和等于蓝数之和.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{592}
Given the function f(x) = sinx, whose graph intersects with the line kx - y - kπ = 0 (k > 0) at exactly three points. The x-coordinates of these points, in ascending order, are x₁, x₂, and x₃. Calculate the value of $$\frac{tan(x_{2} - x_{3})}{x_{1} - x_{2}}$$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
一张台球桌形状是正六边形 $A B C D E F$ ，一个球从 $A B$ 的中点 $P$ 击出，击中 $B C$ 边上的某点 $Q$ ，并且依次碰击 $C D 、 D E 、 E F 、 F A$ 各边，最后击中 $A B$ 边上的某一点. 设 $\angle B P Q=\theta$ ， $\theta$ 的范围为$\theta\in[arc\mathrm{tan}\frac{a\sqrt{3}}{b}, arc\mathrm{tan}\frac{c\sqrt{3}}{d}]$,求$a+b+c+d$。
提示：利用入射角等于反射角的原理。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Two unit squares $S_{1}$ and $S_{2}$ have horizontal and vertical sides. Let $x$ be the minimum distance between a point in $S_{1}$ and a point in $S_{2}$, and let $y$ be the maximum distance between a point in $S_{1}$ and a point in $S_{2}$. Given that $x=5$, the difference between the maximum and minimum possible values for $y$ can be written as $a+b \sqrt{c}$, where $a, b$, and $c$ are integers and $c$ is positive and square-free. Find $100 a+10 b+c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{472}
9. (10 points) The area of rectangle $A B C D$ is 2011 square centimeters. The vertex $F$ of trapezoid $A F G E$ is on $B C$, and $D$ is the midpoint of the non-parallel side $E G$. Find the area of trapezoid $A F G E$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2011}
What is the number of idempotent polynomials in the quotient ring $\Bbb R[x_1,x_2,\dots,x_n]/(x_1^2-x_1,x_2^2-x_2,\dots,x_n^2-x_n)$ that satisfy the relation $p^2=p$ modulo $x_i$ and $x_i-1$ for all $i$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2^{2^n}$}
PQRST is a regular pentagon. The point U lies on ST such that ∠QPU is a right angle. Find the ratio of the interior angles in triangle PUT. Express your answer as a ratio of integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:3:6}
In base $R_1$, the expanded fraction $F_1$ becomes $0.373737\ldots$, and the expanded fraction $F_2$ becomes $0.737373\ldots$. In base $R_2$, fraction $F_1$, when expanded, becomes $0.252525\ldots$, while fraction $F_2$ becomes $0.525252\ldots$. Find the sum of $R_1$ and $R_2$, each written in base ten.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
David draws a $2 \times 2$ grid of squares in chalk on the sidewalk outside NIMO HQ. He then draws one arrow in each square, each pointing in one of the four cardinal directions (north, south, east, west) parallel to the sides of the grid. In how many ways can David draw his arrows such that no two of the arrows are pointing at each other?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{194}
If $\frac{8}{19}$ of the product of the largest two elements of a positive integer set is not greater than the sum of the other elements, what is the minimum possible value of the largest number in the set?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
The positional relationship between the circle $x^2+y^2=4$ and the circle $(x+3)^2+(y-4)^2=16$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{intersect}
In triangle $ABC$, the median from vertex $A$ is perpendicular to the angle bisector of angle $B$, and the median from vertex $B$ is perpendicular to the angle bisector of angle $A$. It is known that side $AB = 1$. Find the perimeter of triangle $ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
A traffic light runs repeatedly through the following cycle: green for \(30\) seconds, then yellow for \(3\) seconds, and then red for \(30\) seconds. Leah picks a random three-second time interval to watch the light. What is the probability, expressed as a fraction in simplest form \(\frac{k}{m}\), that the color changes while she is watching? Please provide the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
How many elements of order $p$ exist in the general linear group $GL_{2}(\Bbb Z_{p})$, where $p$ is a prime number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p^2 - 1}
Let $H = H^1([0,1])$ and let $Tu = u(1)$. Let $\langle \cdot,\cdot \rangle_H$ denote the standard inner product in $H$. By the Riesz representation theorem, there exists a unique $v\in H$ such that $Tu = \langle u, v \rangle_H$ for all $u\in H$. Find $v$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{ \cosh x}{\sinh 1}\)}
Find an explicit formula for a function that is exactly equivalent to the power series $$\sum_{n=1}^\infty\frac{(-1)^n}{2n+1}x^{2n}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\arctan(x)}{x}-1\)}
Given the function $f(x)=\sin (2x+ \frac {π}{3})$, for any $x\_1$, $x\_2$, $x\_3$, and $0\leqslant x\_1 < x\_2 < x\_3\leqslant π$, the equation $|f(x\_1)-f(x\_2)|+|f(x\_2)-f(x\_3)|\leqslant m$ holds true. Find the minimum value of the real number $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3+ \frac { \sqrt {3}}{2}}
The base of the pyramid SABCD is the parallelogram ABCD. Which figure is formed by the section of this pyramid with the plane ABM, where M is a point on the edge SC?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Trapezoid}
If the sines of the internal angles of $\triangle ABC$ form an arithmetic sequence, what is the minimum value of $\cos C$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Let $ABC$ be a right triangle where $\measuredangle A = 90^\circ$ and $M\in (AB)$ such that $\frac{AM}{MB}=3\sqrt{3}-4$. It is known that the symmetric point of $M$with respect to the line $GI$ lies on $AC$. Find the measure of $\measuredangle B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30^\circ}
In a mathematical test for grade 12 students in a certain city, it was found that the scores (X) approximately followed a normal distribution N(84, σ²) after a sample analysis. Also, P(78 < X ≤ 84) = 0.3. If there were 400 participants from a particular school, estimate the number of students from this school who scored no less than 90. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
Mac loves the shape of quarters and is willing to trade money for them, even if he loses value. He tells his brother that he will trade him 4 dimes and 2 pennies for a quarter with a 5% probability or 9 nickels and 1 penny for a quarter with a 10% probability, or he will trade 1 half-dollar coin and 3 cents for a quarter with an 85% probability. Each trade with the given probabilities is repeated 20 times for each option (60 trades in total). How many dollars did Mac lose in these transactions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$5.35}
Russom wants to give some friends envelopes with bus tickets and subway tickets in them. He has 18 bus tickets and some subway tickets to split equally among the envelopes, and wants no tickets left over. The greatest number of envelopes Russom can make is 6. How many subway tickets does he have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
All factors of a positive integer a are multiplied and the product obtained is a ^ 5 . If a is greater than 1, how many unique factors does the expression have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Find the number of prime numbers $p$ between $100$ and $200$ for which the congruence equation $x^{11} + y^{16} \equiv 2013 \pmod{p}$ has a solution in integers $x$ and $y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Let $ABC$ be an equilateral triangle of side length 6 inscribed in a circle $\omega$. Let $A_{1}, A_{2}$ be the points (distinct from $A$) where the lines through $A$ passing through the two trisection points of $BC$ meet $\omega$. Define $B_{1}, B_{2}, C_{1}, C_{2}$ similarly. Given that $A_{1}, A_{2}, B_{1}, B_{2}, C_{1}, C_{2}$ appear on $\omega$ in that order, find the area of hexagon $A_{1}A_{2}B_{1}B_{2}C_{1}C_{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{846\sqrt{3}}{49}}
How many roots does the equation 

$$
\overbrace{f(f(\ldots f}^{10 \text{ times }}(x) \ldots))+\frac{1}{2}=0
$$

where $f(x)=|x|-1$ have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Find the number of strictly increasing sequences of nonnegative integers with the following properties: - The first term is 0 and the last term is 12. In particular, the sequence has at least two terms. - Among any two consecutive terms, exactly one of them is even.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}
在平面直角坐标系 $x O y$ 中, 设椭圆 $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{a^{2}-9}=1(a>3)$.
若直线 $l: \frac{x}{a}+\frac{y}{a-3}=1$ 与椭圆 $C$ 交于 $A, B$ 两点, 其中对任意大于 3 的实数 $a$, 以 $A B$ 为直径的圆过定点, 求定点坐标.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$(-3,0)$}
Given that the weights (in kilograms) of 4 athletes are all integers, and they weighed themselves in pairs for a total of 5 times, obtaining weights of 99, 113, 125, 130, 144 kilograms respectively, and there are two athletes who did not weigh together, determine the weight of the heavier one among these two athletes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{66}
Find the largest \( n \) such that the sum of the fourth powers of any \( n \) prime numbers greater than 10 is divisible by \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{240}
Let \( A, B, C \) be positive integers such that the number \( 1212017ABC \) is divisible by 45. Find the difference between the largest and the smallest possible values of the two-digit number \( AB \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{85}
Milly chooses a positive integer  $n$  and then Uriel colors each integer between  $1$  and  $n$  inclusive red or blue. Then Milly chooses four numbers  $a, b, c, d$  of the same color (there may be repeated numbers). If  $a+b+c= d$  then Milly wins. Determine the smallest  $n$  Milly can choose to ensure victory, no matter how Uriel colors.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Let $\alpha$ and $\beta$ be conjugate complex numbers such that $\frac{\alpha}{\beta^3}$ is a real number and $|\alpha - \beta| = 6$. Find $|\alpha|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\sqrt{2}}
Find a function $f(x)$ that satisfies the following conditions:
- $f(x)$ is twice differentiable on $[0,1]$.
- $f(0) = 2$, $f'(0) = -2$, $f(1) = 1$.
- The differential equation $f''(x) + f(x) f'(x) = 0$ is satisfied for all $x \in (0,1)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{2}{x+1}\)}
Let \( n \) be the largest integer for which \( 14n \) has exactly 100 digits. Counting from right to left, what is the 68th digit of \( n \)? Express your answer as a single digit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the number of triples $0 \leq k, m, n \leq 100$ of integers such that $$ 2^{m} n-2^{n} m=2^{k} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Exercise 18. We say that a set $A$ of polynomials with real coefficients is beautiful if, whenever $P$ and $Q$ are two distinct elements of $A$, there exist positive integers $a_{1}>\cdots>a_{2020}$ such that

$$
P Q=\sum_{i=1}^{2020} i X^{a_{i}}
$$

What is the maximal cardinality of a beautiful set?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the largest positive integer $n$ ($n \ge 3$), such that there exists a convex $n$-gon where the tangent of each interior angle is an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
There are 3 different pairs of shoes in a shoe cabinet. If one shoe is picked at random from the left shoe set of 6 shoes, and then another shoe is picked at random from the right shoe set of 6 shoes, calculate the probability that the two shoes form a pair.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
$36 S$ is a subset of $\{1, 2, \cdots, 1989\}$, and any two numbers in $S$ cannot have a difference of 4 or 7. What is the maximum number of elements that $S$ can have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{905}
A rectangular box has interior dimensions 6-inches by 5-inches by 10-inches. The box is filled with as many solid 3-inch cubes as possible, with all of the cubes entirely inside the rectangular box. What percent of the volume of the box is taken up by the cubes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54\%}
Given the function $f(x)=\frac{-4x+5}{x+1}$ and $g(x)=a\sin(\frac{\pi}{3}x)-2a+2(a>0)$, if there exist $x_{1}$ and $x_{2}\in \left[0,2\right]$ such that $f(x_{1})=g(x_{2})$, then the range of real number $a$ is ______
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(0,3\right]}
Suppose I have a connected graph with $n$ vertices and start at an arbitrary vertex $u$. I want to visit every vertex of the graph, and I don't care about returning to where I started. I can visit a node multiple times if needed. What is the worst possible length of the shortest path, as a function of $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n - 3}
Find the interquartile range (Q3 - Q1) of the following list of numbers: 42, 24, 30, 28, 26, 19, 33, 35, 47, 55, 61, 27, 39, 46, 52, 20, 22, 37, 48, 60, 50, 44, 31, 54, and 58.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23.5}
A random integer $x \in \{1, 2, \cdots, 2016\}$ is equally likely to be generated. Determine the probability that the sum of the digits of $x$ in binary form does not exceed 8.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{655}{672}}