inclusionAI/Ring-lite-rl-data huggingface dataset:
9. A positive integer that can be written in the form $20a + 8b + 27c$ (where $a, b, c$ are non-negative integers) is called a "good number." Then the number of good numbers in the set $\{1, 2, \cdots, 200\}$ is $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{153}
Given that the focus of a parabola is on the y-axis, and the distance from point $M(m,-3)$ on the parabola to the focus is $5$, then $m=$ ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm2\sqrt{6}}
Evaluate the limit for a positive integer $p \geq 2$:

$$ \lim_{n\to\infty}\left(\frac{1}{n+1} + \frac{1}{n+2} + \dots + \frac{1}{pn}\right) $$

Determine the value of this limit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln p}
Let  $S(M)$  denote the sum of the digits of a positive integer  $M$  written in base  $10$ . Let  $N$  be the smallest positive integer such that  $S(N) = 2013$ . What is the value of  $S(5N + 2013)$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
On a long straight section of a two-lane highway where cars travel in both directions, cars all travel at the same speed and obey the safety rule: the distance from the back of the car ahead to the front of the car behind is exactly one car length for every 10 kilometers per hour of speed or fraction thereof. Assuming cars are 5 meters long and can travel at any speed, let $N$ be the maximum whole number of cars that can pass a photoelectric eye placed beside the road in one hour in one direction. Find $N$ divided by $10$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
In the given grid, all circles share the same center. Express the area of the highlighted gray region as a fraction of the area of the largest circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
In the diagram, square PQRS has side length 40. Points J, K, L, and M are on the sides of PQRS, so that JQ = KR = LS = MP = 10. Line segments JZ, KW, LX, and MY are drawn parallel to the diagonals of the square. Find the area of quadrilateral WXYZ. Express your answer in square units.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
Based on the data provided, what were the average daily high and low temperatures in Addington from September 15th, 2008 to September 21st, 2008, inclusive? The high temperatures in degrees Fahrenheit for these days are 49, 62, 58, 57, 46, 55, 60 and the corresponding low temperatures are 40, 47, 45, 41, 39, 42, 44. Express your answer as a decimal to the nearest tenth.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42.6}
Let  $S$  be a finite set of points in the plane. A linear partition of  $S$  is an unordered pair  $\{A,B\}$  of subsets  of  $S$  such that  $A\cup B=S,\ A\cap B=\emptyset,$  and  $A$  and  $B$  lie on opposite sides of some straight line disjoint from  $S$  ( $A$  or  $B$  may be empty). Let  $L_{S}$  be the number of linear partitions of  $S.$  For each positive integer  $n,$  find the maximum of  $L_{S}$  over all sets  $S$  of  $n$  points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n(n-1)}{2} + 1}
Calculate the value of the limit: $$ \lim _{n \rightarrow \infty} \frac{96}{n^4}\left[1\left(\sum_{k=1}^n k\right)+2\left(\sum_{k=1}^{n-1} k\right)+3\left(\sum_{k=1}^{n-2} k\right)+\cdots+n \cdot 1\right] $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Evaluate the contour integral \( \displaystyle\int_{|z|=4} \frac {z} {\sin z}\,\mathrm dz \), given that \( \sin z \) has three zeros within the circle \( |z|=4 \), making the integrand non-analytic at those points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the remainder when 25! is divided by 78125.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15625}
Evaluate the limit: $$\lim_{n \to \infty} \int_{\frac{1}{(n+1)^2}}^{\frac{1}{n^2}} \frac{e^x\sin^2(x)}{x^{7/2}}\, \mathrm{d}x$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the number of zeroes of the function \( f(z) = z^4 + e^z \) within the unit disc \( |z| < 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
马休先按顺序写出了1到10000的全部数字，然后擦去了那些既不能被5整除，又不能被11整除的数，在剩下的数中，位于第2004位的数是多少？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7348}
Use the divergence theorem to compute the flux integral \( \iint_S F \cdot dS \), where \( F(x,y,z) = yj - zk \) and \( S \) consists of the union of the paraboloid \( y = x^2 + z^2, 0 \leq y \leq 1 \), and the disk \( x^2 + z^2 \leq 1, y = 1 \), oriented outward. What is the flux through just the paraboloid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\pi}
If the complex number $z$ and $(z + 2)^2 - 8i$ are both purely imaginary, then $z = \boxed{\text{\_\_\_\_\_\_\_\_}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2i}
How many lattice points $(v, w, x, y, z)$ does a $5$-sphere centered on the origin, with radius $3$, contain on its surface or in its interior?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1343}
Evaluate the real convergent improper integral \(\int_0^\infty \frac{\sqrt{x}}{x^3+1}\ \mathrm{d}x\) using residue theory with a keyhole contour that excludes the branch cut along the positive real axis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{3}}
Evaluate the integral \( \frac{1}{2i\pi} \oint_\gamma \frac{2i}{z^2 +1}dz \) where \( \gamma \) is a closed and continuously differentiable path in the upper half-plane \( \{z \in \mathbb{C} : z = x + iy, x, y \in \mathbb{R}, y > 0\} \) that does not pass through the point \( i \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the radius of convergence of the series: $$1+ \frac{a.b}{1.c}z + \frac{a(a+1)b(b+1)}{1.2...c(c+1)}z^2+.... $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Initially, one of the two boxes on the table is empty and the other contains 29 different colored marbles. Starting with the full box, in each move, one or more marbles are selected from that box and transferred to the other box. What is the maximum number of moves that can be made without selecting the same set of marbles more than once?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{29} - 1}
A tetrahedron with four equilateral triangular faces has a sphere inscribed within it and a sphere circumscribed about it. For each of the four faces, there is a sphere tangent externally to the face at its center and to the circumscribed sphere. A point $ P$ is selected at random inside the circumscribed sphere. The probability that $ P$ lies inside one of the five small spheres is closest to
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.2}
Find the kernel of the matrix \(A\) defined as \[A:=\begin{pmatrix} x_1^0 & x_1^1 & \ldots & x_1^n\\ x_2^0 & x_2^1 & \ldots & x_2^n\\ \vdots & \vdots & \ddots & \vdots\\ x_m^0 & x_m^1 & \ldots & x_m^n \end{pmatrix}\] where \(n < m - 1\) and the elements \(x_i\) are distinct.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{\mathbf{0}\}}
Given two independent random variables, where \(X\) follows a uniform distribution \(U(0,\pi)\) and \(Y\) follows a uniform distribution \(U(0,2\pi)\), calculate the probability \(\Pr(Y \in [\pi, X+\pi] \mid X)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{X}{2\pi}}
In the rectangular coordinate plane $xOy$, two circles are given:
$\odot O: x^{2}+y^{2}=1$,
$\odot O_{1}:(x-3)^{2}+y^{2}=4$.

From a point $M$ on the left half of the $x$-axis, a tangent is drawn to $\odot O$, touching $\odot O$ at point $A$ and intersecting $\odot O_{1}$ at points $B$ and $C$. Given that $AB = BC$, find the coordinates of point $M$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-4, 0)}
设 $x \neq y$, 且两数列 $x, a_{1}, a_{2}, a_{3}, y$ 和 $b_{1}, x, b_{2}, b_{3}, y, b_{4}$ 均为等差数列, 那么 $\frac{b_{4}-b_{3}}{a_{2}-a_{1}}=$ $\qquad$.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{11}
Determine the VC-dimension of the class $H_{N-parity}$ consisting of all parity functions $h_I: \{0,1\}^N \rightarrow \{0,1\}$, where $I \subseteq \{1,2,..., N\}$, defined by $h_I(x) = \bigg(\sum_{i\in I}x_i\bigg) \mod 2$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{N}
Calculate the derivative of the function \( F(x) = \int_0^2 \sin(x+l)^2\ dl \) with respect to \( x \) at \( x=0 \). Use the Leibniz rule to differentiate \( F(x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sin(4)}
Let \( f(x, y) \) be a homogeneous function of degree 3. Given that \( f'_x(2, 1) = 1 \), \( f(1, 0.5) = 2 \), \( f(4, 2) = 128 \), and \( f(2, 1) = 16 \), compute \( f'_y(4, 2) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{184}
在已知数列1,4,8,10,16,19,21,25,30,43中，相邻若干数之和能被11整除的数组共有＿组．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
A regular hexagon has side length \(6\). Congruent arcs with radius \(3\) are drawn with the center at each of the vertices, creating circular sectors as shown. The region inside the hexagon but outside the sectors is shaded as shown. The original answer is in the form \(a\sqrt{b}-c\pi\). Please find the value of a + b + c.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}
从0、2、4中取一个数字，从1、3、5中取两个数字，组成无重复数字的三位数，则所有不同的三位数的个数是______（用数字作答）
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$48$}
A family consists of 5 brother-sister pairs. You have 10 identical Barbie dolls and 10 identical toy cars to distribute among the sisters and brothers, respectively, ensuring that each child gets a gift. Additionally, each brother-sister pair must receive at least 3 gifts in total. Calculate the number of ways to distribute the gifts.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4251}
Select 2 elements from each of the sets {O, P, Q, R, S} and {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} to form a sequence (both letters and numbers cannot be repeated). The number of different sequences where at most one of the letters O, Q, and the numbers 0 can appear is ___ . (Answer with a number).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8424}
Let the set \( A = \{1, 2, \cdots, 2016\} \). A 1008-element subset \( X \) of \( A \) is called a "good set" if there exist \( x, y \in X \) such that \( x < y \) and \( x \mid y \). Determine the largest positive integer \( a \in A \) such that every 1008-element subset containing \( a \) is a "good set".
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{672}
Evaluate the integral:

$$\int_{-\infty}^\infty\frac{2021^{-|x|}}{1+3^{\arctan(\tan^3(x))}}dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{1}{\ln(2021)}\)}
4. A cylinder is intersected by a plane, and the shape of the intersection is an ellipse. The eccentricity of this ellipse is $\frac{\sqrt{5}}{3}$. The acute angle formed between this intersection and the base of the cylinder is . $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\arccos\frac{2}{3}}
Two triangular pyramids \( MABC \) and \( NABC \) have a common base and no other common points. All vertices of both pyramids lie on the same sphere. Find the lengths of the edges \( MA \) and \( MB \) if it is known that they are equal to each other, and the lengths of all other edges of these pyramids are \( \sqrt{3} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A $7 \times 7$ grid is colored black and white. If the number of columns with fewer black squares than white squares is $m$, and the number of rows with more black squares than white squares is $n$, find the maximum value of $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Dr. Banks had some toothbrushes to give away to his patients. He gave away 53 toothbrushes in January, 67 toothbrushes in February, 46 toothbrushes in March, and the remaining toothbrushes were given away half each in April and May. The difference between the busiest and slowest month was 36 toothbrushes. How many toothbrushes did Dr. Banks have initially?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{228}
In the diagram, the area of square \( QRST \) is 36. Also, the length of \( PQ \) is one-half of the length of \( QR \). What is the perimeter of rectangle \( PRSU \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
In the given equation, each letter represents a number between 1 and 9, with different letters representing different digits. Find the value of $A+B+C=$ $\qquad$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Consider the number obtained by writing the numbers  $1,2,\ldots,1990$  one after another. In this number every digit on an even position is omitted; in the so obtained number, every digit on an odd position is omitted; then in the new number every digit on an even position is omitted, and so on. What will be the last remaining digit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Find the Fourier transform of the product $f(x)\delta(x)$, where $f(x)$ is a continuous function and $\delta(x)$ is the Dirac delta function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{f(0)}{\sqrt{2\pi}}}
Out of the 800 emails that Antonia received in her mail, 3/7 were spam emails, while 5/11 of the remaining emails were promotional messages. If the rest of the emails were important emails, calculate the total number of important emails in her inbox.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{249}
Consider the function \( f(x) = x\sqrt{x\sqrt{x\sqrt{x...\infty}}} \). If \( f(x) \) is defined as the limit of the expression as the number of roots approaches infinity, determine if \( f'(5) \) exists. If it does, find its value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Find the number of solutions to the equation $\log_e |2\cos x| = - x^2 + x$ in the interval $[0, 2\pi]$. Do not use a graphing calculator.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the order of growth for the entire function represented by the infinite product \( f(z) = \prod_{n=1}^{\infty} \left(1-\frac{z}{n!}\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the condition on the $L^2$ norm of the Hilbert-Schmidt kernel $k$ that ensures the invertibility of the operator $I-T$, where $T$ is an integral operator with kernel $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\|k\|_2 < 1}
Find the limit \( \lim_{n\rightarrow\infty}\int_{0}^{1}nx^{n}f(x)dx \) where \( f(x) \) is a Lebesgue integrable function on \([0,1]\) satisfying \( \lim_{x\rightarrow1^-}f(x) = \xi \) for some finite \( \xi \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\xi}
Find the center of the group with the presentation \( \langle a, b \mid a^2 = b^3 \rangle \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\langle a^2 \rangle}
What is the cardinality of the set of all infinite monotonically non-increasing sequences of natural numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\aleph_0}
Each cell of a $100 \times 100$ board is painted in either blue or white. We call a cell balanced if it has an equal number of blue and white neighboring cells. What is the maximum number of balanced cells that can be found on the board? (Cells are considered neighbors if they share a side.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9608}
In an isosceles trapezoid with an acute angle \(\alpha\) at the base, a circle constructed on one of the legs as its diameter is tangent to the other leg.

In what ratio does it divide the larger base of the trapezoid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sin 2 \alpha : 1}
Let  $f(x)=cx(x-1)$ , where  $c$  is a positive real number.  We use  $f^n(x)$  to denote the polynomial obtained by composing  $f$  with itself  $n$  times.  For every positive integer  $n$ , all the roots of  $f^n(x)$  are real.  What is the smallest possible value of  $c$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A swimming pool is 20 ft wide, 40 ft long, and has a variable depth: 3 ft at the shallow end and 9 ft at the deepest point. Water is being filled at a rate of 0.7 ft³/min. Determine how fast the water level is rising when the depth at the deepest point is 5 ft. Express your answer rounded to five decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.00105}
Circles $C_1$ and $C_2$ intersect at points $X$ and $Y$ . Point $A$ is a point on $C_1$ such that the tangent line with respect to $C_1$ passing through $A$ intersects $C_2$ at $B$ and $C$, with $A$ closer to $B$ than $C$, such that $2016 \cdot AB = BC$. Line $XY$ intersects line $AC$ at $D$. If circles $C_1$ and $C_2$ have radii of $20$ and $16$, respectively, find $\sqrt{1+BC/BD}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2017}
Let $p(x)=c_1+c_2\cdot2^x+c_3\cdot3^x+c_4\cdot5^x+c_5\cdot8^x$. Given that $p(k)=k$ for $k=1,2,3,4,5$, compute $p(6)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-50}
Find the minimum value of  $c$  such that for any positive integer  $n\ge 4$  and any set  $A\subseteq \{1,2,\cdots,n\}$ , if  $|A| >cn$ , there exists a function  $f:A\to\{1,-1\}$  satisfying  $$ \left| \sum_{a\in A}a\cdot f(a)\right| \le 1. $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{3}}
Determine the minimum number of weighings necessary to identify two groups of 4 boxes each, such that the total weight of the boxes in each group is equal, using a balance scale with two pans.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Identify a function for which the Taylor series expansion does not represent the function within an interval containing the expansion point.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-1/x^2}}
Determine the value of the limit \( L = \lim \limits_{x \to 0}\frac{a-\sqrt{a^2-x^2}-\frac{x^2}{4}}{x^4} \) such that \( L \) is finite. What is the value of \( a \) that satisfies this condition?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the rank of the linear operator on the vector space of real 3x3 matrices that maps a matrix X to AX - XA, where \( A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{pmatrix} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the maximum possible dimension of a residue field of the polynomial ring $k[x_1, x_2, \ldots]$ over an uncountable field $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\aleph_0}
Calculate the volume of the solid obtained by rotating the region bounded by the lines $x=2$, $y=3$, and the curves $xy=2$, $xy=4$ around the x-axis using the method of cylindrical shells.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6\pi}
Find the limit of the sequence \(\{x_n\}\) as \(n\) approaches infinity, where \(x_n\) satisfies the equation \(x_n^3 - (n^2 + 3n)x_n - 3n^2 = 0\) for all \(n \in \mathbb{N}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3}
Consider the polynomial \( \prod_{n = 1}^{2023}\left( 1 + nx^{5^{n}}\right) = \sum_{n=0}^{m}a_{n}\,x^{k_{n}} \) where the \(k_{n}\) are in increasing order, and the \(a_{n}\) are nonzero. Determine the coefficient \(a_{2023}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1995840}
Evaluate the integral \( I = \int_{-\infty}^{\infty} \frac{e^{\cos(x)} \sin(\sin(x))}{x} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi(e - 1)}
In a semicircle with diameter $AB$ and center $S$, points $C$ and $D$ are located such that $C$ is on arc $AD$ and $\angle CSD = 120^\circ$. Let $E$ be the intersection of lines $AC$ and $BD$, and $F$ be the intersection of lines $AD$ and $BC$. Determine the value of $\frac{EF}{AB}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
Evaluate the limit as \( n \to \infty \):
\[ \lim_{n \to \infty} \left[1- p(n) \right]^{n-1} \]
where \( p(n) = o \left( \frac{\log n}{n} \right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A plane passes through the vertex \( A \) of a right circular cone, creating a cross-section of maximum area. This area is twice the area of a cross-section that passes through the cone's axis. Determine the angle at the vertex of the cone's axial cross-section.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150^\circ}
Given the equation \(2\tan^2x - 5\sec x = 1\), which has exactly 7 distinct solutions for \(x\in[0,\frac{n\pi}{2}]\), where \(n\in\mathbb{N}\), find the greatest possible value of \(n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Determine the number of solutions of the polynomial equation \(z^4 - 7z^3 - 2z^2 + z - 3 = 0\) that lie inside the unit disc.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Identify a non-commutative ring \( B \) such that its only left ideals are \((0)\) and \( B \) itself.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{H}}
Determine the values of \(a\) for which the function \(f(x,y) = -9x^2 + axy - y^2 + 4ay\) is convex everywhere and in all directions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
From the $8$ vertices of a cube, choose any $4$ vertices. The probability that these $4$ points lie in the same plane is ______ (express the result as a simplified fraction).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{6}{35}}
Evaluate or approximate the integral \( \int_{-\frac{b}{2}}^{\frac{b}{2}} \ln(f(x)) \, dx \) for the function \( f(x) = 350 + 5\left(1 - \frac{\cosh\left(\frac{x}{4 \cdot 10^{-6}}\right)}{\cosh\left(\frac{10^{-4}}{8 \cdot 10^{-6}}\right)}\right) \) over the interval \(-\frac{10^{-4}}{2} < x < \frac{10^{-4}}{2}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5.87 \times 10^{-4}}