inclusionAI/Ring-lite-rl-data huggingface dataset:
In the expansion of the binomial $({1 \over 2} + 2x)^n$, the sum of the binomial coefficients of the first three terms is equal to $79$. Determine the coefficient of $x^4$ in the expansion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{{495 \over 16}}
Let $S = \{ 1, 2, \ldots, 15 \}$. Let $A_1, A_2, \ldots, A_n$ be $n$ subsets of $S$ which satisfy the following conditions:

1. $|A_i| = 7$, for $i = 1, 2, \ldots, n$;

2. $|A_i \cap A_j| \leq 3$, for $1 \leq i < j \leq n$;

3. For any 3-element subset $M$ of $S$, there exists $A_k$ such that $M \subset A_k$.

Find the smallest possible value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
A square sheet of paper has an area of $12 \text{ cm}^2$. The front is white and the back is black. When the paper is folded so that point $A$ rests on the diagonal and the visible black area is equal to the visible white area, how far is point A from its original position? Give your answer in simplest radical form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{6}}
Through the points \( A(0 ; 14) \) and \( B(0 ; 4) \), two parallel lines are drawn. The first line, passing through point \( A \), intersects the hyperbola \( y = \frac{1}{x} \) at points \( K \) and \( L \). The second line, passing through point \( B \), intersects the hyperbola \( y = \frac{1}{x} \) at points \( M \) and \( N \).

What is the value of \( \frac{A L-A K}{B N-B M} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.5}
设平面点集 $A=\left\{(x, y) \left\lvert\,(y-x) \cdot\left(y-\frac{18}{25 x}\right) \geqslant 0\right.\right\}, B=\left\{(x, y) \mid(x-1)^2+(y-1)^2 \leqslant 1\right\}$. 若 $(x, y) \in A \cap B$, 求 $2 x-y$的最小值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
In triangle \( \triangle ABC \), given \( \angle A = \alpha \), and \( CD \) and \( BE \) are the altitudes on \( AB \) and \( AC \) respectively, find \( \frac{DE}{BC} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|\cos \alpha|}
Let  $n$  be positive integer and  $S$ = { $0,1,…,n$ }, Define set of point in the plane.   $$ A = \{(x,y) \in S \times S \mid -1 \leq x-y \leq 1 \}  $$  , We want to place a electricity post on a point in  $A$  such that each electricity post can shine in radius 1.01 unit. Define minimum number of electricity post such that every point in  $A$  is in shine area
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ n+1 }
On a long bench, a boy and a girl were sitting. One by one, 20 more children approached them, and each sat between two children already sitting there. We call a girl "brave" if she sat between two neighboring boys, and a boy "brave" if he sat between two neighboring girls. When everyone was seated, it turned out that boys and girls were sitting on the bench alternately. How many of them were brave?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Given the ellipse \( C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) where \( a > b > 0 \), let \( F \) be the right focal point of the ellipse \( C \). A line \( l \) is drawn through the intersection of the right directrix \( x = 2a \) and the \( x \)-axis, and intersects the ellipse \( C \) at points \( A \) and \( B \). The arithmetic mean of \( \frac{1}{|AF|} \) and \( \frac{1}{|BF|} \) is \( \frac{1}{24} \). Find the maximum area of triangle \( \triangle ABF \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{192 \sqrt{3}}
Lil writes one of the letters P, Q, R, S in each cell of a 2x4 table. She does this in such a way that, in each row and in each 2x2 square, all four letters appear. How many distinct arrangements can she make? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Given the function $y=f(x)$, let $A(x\_1,y\_1)$ and $B(x\_2,y\_2)$ be two distinct points on its graph with the slopes of the tangent lines at these points being $k\_A$ and $k\_B$, respectively. Define $K(A,B)=\frac{|k\_A-k\_B|}{|AB|}$ (where $|AB|$ denotes the length of the line segment $AB$) as the "approximate curvature" of the curve $y=f(x)$ between points $A$ and $B$. For the curve $y=\frac{1}{x}$, consider two points $A(a,\frac{1}{a})$ and $B(\frac{1}{a},a)$ with $a>0$ and $a\neq 1$. Determine the range of real values of $m$ such that $m\cdot K(A,B)>1$ always holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[\frac{\sqrt{2}}{2},+\infty)}
A positive integer $n > 1$ is called beautiful if $n$ can be written in one and only one way as $n = a_1 + a_2 + \cdots + a_k = a_1 \cdot a_2 \cdots a_k$ for some positive integers $a_1, a_2, \ldots, a_k$, where $k > 1$ and $a_1 \geq a_2 \geq \cdots \geq a_k$. For example, 6 is beautiful since $6 = 3 \cdot 2 \cdot 1 = 3 + 2 + 1$, and this is unique. But 8 is not beautiful since $8 = 4 + 2 + 1 + 1 = 4 \cdot 2 \cdot 1 \cdot 1$ as well as $8 = 2 + 2 + 2 + 1 + 1 = 2 \cdot 2 \cdot 2 \cdot 1 \cdot 1$, so uniqueness is lost. Find the largest beautiful number less than 100.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{95}
Let $S_P$ be the set of all polynomials $P$ with complex coefficients, such that $P(x^2) = P(x)P(x-1)$ for all complex numbers $x$. Suppose $P_0$ is the polynomial in $S_P$ of maximal degree such that $P_0(1) \mid 2016$. Find $P_0(10)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12321}
A plane contains points $A$ and $B$ with $AB = 1$. Let $S$ be the union of all disks of radius $1$ in the plane that cover $\overline{AB}$. What is the area of $S$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\pi - \dfrac{\sqrt{3}}{2}}
Let \(T_1\) be a triangle with side lengths \(2011, 2012,\) and \(2013\). For \(n \ge 1\), if \(T_n = \triangle ABC\) and \(D, E,\) and \(F\) are the points of tangency of the incircle of \(\triangle ABC\) to the sides \(AB, BC\), and \(AC,\) respectively, then \(T_{n+1}\) is a triangle with side lengths \(AD, BE,\) and \(CF,\) if it exists. Find the perimeter of the last triangle in the sequence \(( T_n )\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1637}
You invest a certain amount in two accounts paying 8% and 5% respectively. At the end of the year, the interest is $490. You invested $3000 at one rate and the remaining amount at the other rate. How much money did you invest in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$8000}
We play on a chessboard with sides of length 100. A move consists of choosing a rectangle formed by squares on the chessboard and inverting the colors. What is the smallest number of moves needed to make the entire chessboard completely black?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
将一个色子连续投掷五次，则事件 "五次出现的点数既不全相同也不两两互异，且从第二次起每一次的点数都不小于前一次的点数"的概率是 $\qquad$.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{167}
In $\triangle ABC$, if $b=2\sqrt{2}$, $a=2$, and the triangle has a solution, then the range of values for $A$ is ________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0^\circ, 45^\circ]}
A counter is placed at position 0 on a circle with 10 points (indexed 0 through 9). On the nth move, the counter moves n^n steps clockwise. Find the position of the counter after 1234 moves. Express your answer as a single integer representing the final position.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
A Gauss brand light bulb will work for 24999 hours. If it is used for exactly 2 hours every day starting on a Monday, on what day of the week will it stop working? Express your answer as a day of the week (e.g. Monday, Tuesday, etc.).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Friday}
There are nine lesson preparation groups in the senior grade three, with a minimum of four people per group. 12 individuals are now selected from these groups, with at least one person from each group, to form the "core group of the grade" to discuss relevant matters. Calculate the total number of different allocation schemes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{165}
Let $P$ be a point on the circumcircle of square $ABCD$. Find all integers $n > 0$ such that the sum $$S^n(P) = PA^n + PB^n + PC^n + PD^n$$ is constant with respect to point $P$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
If the fractional equation $\frac{3}{{x-4}}+\frac{{x+m}}{{4-x}}=1$ has a root, determine the value of $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Find the value of $\int_0^3 (x^2 + 1){\rm d}\left[x\right]$ where [.] denotes the greatest integer function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
有六面旗, 两面蓝, 两面红, 两面黄, 除颜色外完全相同, 从这些旗子中去除若干面（至少一面），从上到下悬挂在同一个旗杆上，可以组成一个信号序列，则不同的信号序列共有多少种?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{180}
$A, B, C, D$ attended a meeting, and each of them received the same positive integer. Each person made three statements about this integer, with at least one statement being true and at least one being false. Their statements are as follows:

$A:\left(A_{1}\right)$ The number is less than 12;
$\left(A_{2}\right)$ 7 cannot divide the number exactly;
$\left(A_{3}\right)$ 5 times the number is less than 70.

$B:\left(B_{1}\right)$ 12 times the number is greater than 1000;
$\left(B_{2}\right)$ 10 can divide the number exactly;
$\left(B_{3}\right)$ The number is greater than 100.

$C:\left(C_{1}\right)$ 4 can divide the number exactly;
$\left(C_{2}\right)$ 11 times the number is less than 1000;
$\left(C_{3}\right)$ 9 can divide the number exactly.

$D:\left(D_{1}\right)$ The number is less than 20;
$\left(D_{2}\right)$ The number is a prime number;
$\left(D_{3}\right)$ 7 can divide the number exactly.

What is the number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{89}
The sum of the first thirteen terms of an arithmetic progression is $50\%$ of the sum of the last thirteen terms of this progression. The sum of all terms of this progression without the first three terms is related to the sum of all terms without the last three terms as $3:2$. Find the number of terms in this progression.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{{18}.}
In a reunion, there are 20 boys. Out of these, 15 boys are wearing a blue shirt. Each boy shakes hands exactly once with each of the others wearing the same color shirt. What is the total number of handshakes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{105}
8. Let $S$ be the set of discs $D$ contained completely in the set $\{(x, y): y<0\}$ (the region below the $x$-axis) and centered (at some point) on the curve $y=x^{2}-\frac{3}{4}$. What is the area of the union of the elements of $S$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2\pi}{3}+\frac{\sqrt{3}}{4}}
In base \( R_1 \), the fractional expansion of \( F_1 \) is \( 0.373737 \cdots \), and the fractional expansion of \( F_2 \) is \( 0.737373 \cdots \). In base \( R_2 \), the fractional expansion of \( F_1 \) is \( 0.252525 \cdots \), and the fractional expansion of \( F_2 \) is \( 0.525252 \cdots \). What is the sum of \( R_1 \) and \( R_2 \) (both expressed in decimal)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
Problem
Steve is piling $m\geq 1$ indistinguishable stones on the squares of an $n\times n$ grid. Each square can have an arbitrarily high pile of stones. After he finished piling his stones in some manner, he can then perform stone moves, defined as follows. Consider any four grid squares, which are corners of a rectangle, i.e. in positions $(i, k), (i, l), (j, k), (j, l)$ for some $1\leq i, j, k, l\leq n$ , such that $i<j$ and $k<l$ . A stone move consists of either removing one stone from each of $(i, k)$ and $(j, l)$ and moving them to $(i, l)$ and $(j, k)$ respectively,j or removing one stone from each of $(i, l)$ and $(j, k)$ and moving them to $(i, k)$ and $(j, l)$ respectively.
Two ways of piling the stones are equivalent if they can be obtained from one another by a sequence of stone moves.
How many different non-equivalent ways can Steve pile the stones on the grid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\[
\binom{n+m-1}{m}^{2}
\]}
Find the volume $V$ of the solid inside both $x^{2}+y^{2}+z^{2}=4$ and $x^{2}+y^{2}=1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{4 \pi}{3}\left(8-3^{3 / 2}\right)$}
Square \( A B C D \) has sides of length \( 3 \text{ cm} \). The points \( M \) and \( N \) lie on \( A D \) and \( A B \) so that \( C M \) and \( C N \) split the square into three pieces of the same area. Find the length of \( D M \). Express your answer in centimeters (cm).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \text{ cm}}
称形如 $\{x, 2 x, 3 x\}$ 的集合为"好的"。对于给定的整数 $n(n \geqslant 3)$，问：由 $n$ 个正整数构成的集合最多能有多少个好子集？请提供当 n = 10 时的答案。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
For $k \in \{ 0, 1, \ldots, 9 \}$, let $\epsilon_k \in \{-1, 1 \}$. If the minimum possible value of $\sum_{i = 1}^9 \sum_{j = 0}^{i-1} \epsilon_i \epsilon_j 2^{i+j}$ is $m$, find $|m|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{174762}
A 6x6x6 cube is formed by assembling 216 unit cubes. Two 1x6 stripes are painted on each of the six faces of the cube parallel to the edges, with one stripe along the top edge and one along the bottom edge of each face. How many of the 216 unit cubes have no paint on them?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}
Put 8 different balls into three boxes numbered 1, 2, and 3, ensuring that the number of balls in each box is not less than the box's number. There are a total of $\_\_\_\_\_\_\_\_$ different ways to place the balls (answer in digits).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2268}
In a class exam, there are 3 multiple-choice questions, each with 4 possible answer choices. It is known that a group of students' answers have the property $P$: any two students' answers have at most 1 question with the same answer. Furthermore, if one more student is added to this group, regardless of how this student answers, property $P$ no longer holds. What is the minimum number of students in this group?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given a set of data: $24$, $30$, $40$, $44$, $48$, $52$. The average of the $30$th percentile and the $50$th percentile of this data set is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Given the function $f(x)=ax^{2}+bx+c(a,b,c∈R)$, if there exists a real number $a∈[1,2]$, for any $x∈[1,2]$, such that $f(x)≤slant 1$, then the maximum value of $7b+5c$ is _____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-6}
Given the sequence

$$
a_{n}=\frac{(n+3)^{2}+3}{n(n+1)(n+2)} \cdot \frac{1}{2^{n+1}}
$$

defined by its general term, form the sequence

$$
b_{n}=\sum_{k=1}^{n} a_{k}
$$

Determine the limit of the sequence $b_{n}$ as $n$ approaches $+\infty$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
# Problem №3

A New Year's garland hanging along the school corridor consists of red and blue bulbs. Next to each red bulb, there is definitely a blue one. What is the maximum number of red bulbs that can be in this garland if there are 50 bulbs in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
A frog sitting at the point \((1, 2)\) begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length \(1\), and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices \((0,0), (0,4), (4,4),\) and \((4,0)\). What is the probability that the sequence of jumps ends on a vertical side of the square? The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
What is the least positive integer  $k$  such that, in every convex 1001-gon, the sum of any k diagonals is greater than or equal to the sum of the remaining diagonals?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{249750}
Regular hexagon $P_{1} P_{2} P_{3} P_{4} P_{5} P_{6}$ has side length 2. For $1 \leq i \leq 6$, let $C_{i}$ be a unit circle centered at $P_{i}$ and $\ell_{i}$ be one of the internal common tangents of $C_{i}$ and $C_{i+2}$, where $C_{7}=C_{1}$ and $C_{8}=C_{2}$. Assume that the lines $\{\ell_{1}, \ell_{2}, \ell_{3}, \ell_{4}, \ell_{5}, \ell_{6}\}$ bound a regular hexagon. The area of this hexagon can be expressed as $\sqrt{\frac{a}{b}}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1603}
Two counterfeit coins of equal weight are mixed with $8$ identical genuine coins. The weight of each of the counterfeit coins is different from the weight of each of the genuine coins. A pair of coins is selected at random without replacement from the $10$ coins. A second pair is selected at random without replacement from the remaining $8$ coins. The combined weight of the first pair is equal to the combined weight of the second pair. What is the probability that all $4$ selected coins are genuine?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$ \frac{15}{19}$
}
The bases \(AB\) and \(CD\) of the trapezoid \(ABCD\) are 367 and 6 respectively, and its diagonals are mutually perpendicular. Find the scalar product of the vectors \(\overrightarrow{AD}\) and \(\overrightarrow{BC}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2202}
In the cube ABCD-A1B1C1D1, P is a moving point on the side BB1C1C. If the distance from P to line BC and line C1D1 is equal, then what is the curve of the trajectory of the moving point P? Express your answer as the name of a curve.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\text{Parabola}}
Jennifer had $90 to spend on herself. She spent 1/5 of the money on a sandwich, 1/6 for a ticket to a museum, and 1/2 of it on a book. Later, she received an additional $30 as a gift. She then spent 1/4 of her remaining money on cosmetics and 1/3 of what was left on a T-shirt. The sandwich shop had a 10% discount, and there was a 5% sales tax on the T-shirt. How much money does Jennifer have left over?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$21.35}
Given the infinite sequence $\{a_n\}$, where $a_1=1$, $a_2=2$, and for any $n\in\mathbb{N}^*$, it holds that $a_{n+2}=a_n$. The sequence $\{b_n\}$ satisfies $b_{n+1}-b_n=a_n$ ($n\in\mathbb{N}^*$). If any term in the sequence $\left\{\frac{b_{2n}}{a_n}\right\}$ repeats infinitely many times within the sequence, then the value of $b_1$ that meets the requirement is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find an odd natural number not exceeding $1000$ such that the sum of the last digits of all its divisors (including $1$ and the number itself) is $33$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{729}
It is known that the equations \(x^{2}+(2a-5)x+a^{2}+1=0\) and \(x^{3}+(2a-5)x^{2}+(a^{2}+1)x+a^{2}-4=0\) have common roots. Find the sum of these roots.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Given that the area of triangle \( \triangle ABC \) is 10, let \( D, E, F \) be points on \( AB, BC, AC \) respectively, different from \( A, B, C \), with \( AD = 2 \) and \( DB = 3 \). If the areas of \( \triangle ABE \) and quadrilateral \( DBEF \) are equal, calculate the area of \( \triangle ABE \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let \( S = \{1, 2, \ldots, n\} \). \( A \) is an arithmetic sequence with at least two terms and a positive common difference, with all its terms in \( S \). Adding any other element from \( S \) to \( A \) does not form another arithmetic sequence with the same common difference as \( A \). Determine the number of such sequences \( A \). (An arithmetic sequence with only two terms is also considered an arithmetic sequence).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left\lfloor \frac{n^2}{4} \right\rfloor}
Let $P$ be the product of all the entries in row 2021 of Pascal's triangle (the row that begins 1, 2021, \ldots). What is the largest integer $j$ such that $P$ is divisible by $101^j$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1980}
In 1991 the population of a town was a perfect square. Ten years later, after an increase of 150 people, the population was 9 more than a perfect square. Now, in 2011, with an increase of another 150 people, the population is once again a perfect square. What is the percent growth of the town's population during this twenty-year period?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{62}
The height of the isosceles trapezoid $ABCD$ with bases $AD$ and $BC$ is $4 \sqrt{3}$. The diagonals of the trapezoid intersect at point $O$, and $\angle AOD = 120^{\circ}$. Find the median of the trapezoid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Use the residue theorem to compute $$\sum_{k=1}^n\frac{1}{\sin^2\left(\frac{(2k-1)\pi}{2n}\right)}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^2}
A group of n  friends wrote a math contest consisting of eight short-answer problem  $S_1, S_2, S_3, S_4, S_5, S_6, S_7, S_8$ , and four full-solution problems  $F_1, F_2, F_3, F_4$ . Each person in the group correctly solved exactly 11 of the  12 problems. We create an 8 x 4 table. Inside the square located in the  $i$ th row and  $j$ th column, we write down the number of people who correctly solved both problem  $S_i$  and  $F_j$ . If the 32 entries in the table sum to 256, what is the value of n?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Given a sequence \(\left\{a_{n}\right\}\) with the general term
$$
a_{n}=2^{n}+3^{n}+6^{n}-1 \quad (n \in \mathbf{Z}_{+}),
$$
find the positive integers that are coprime with every term of this sequence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A $3 \times 3 \times 3$ cube composed of $27$ unit cubes rests on a horizontal plane. Determine the number of ways to select two distinct unit cubes from a $3 \times 3 \times 1$ block (the order is irrelevant) such that the line joining the centers of the two cubes forms a $45^\circ$ angle with the horizontal plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}