inclusionAI/Ring-lite-rl-data huggingface dataset:
Find the greatest positive integer $A$ with the following property: For every permutation of \{1001, 1002, \ldots, 2000\}, the sum of some ten consecutive terms is greater than or equal to $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15005}
In square \(ABCD\) with side length 9, point \(P\) is on \(AB\) such that \(AP: PB = 7:2\). A quarter circle is drawn inside the square with center \(C\) and radius \(CB\). The tangent from \(P\) meets the circle at \(E\) and \(AD\) at \(Q\). Segments \(CE\) and \(DB\) intersect at \(K\), and \(AK\) and \(PQ\) intersect at \(M\). Determine the length of \(AM\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{85}{22}}
Suppose we have $n$ independent normal random variables $X_1, X_2, \ldots, X_n$ and another independent normal random variable $Y$. What is the probability that $Y$ is less than all of $X_1, X_2, \ldots, X_n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{n+1}}
Find the greatest $a$ such that $\frac{7\sqrt{(2a)^2+(1)^2}-4a^2-1}{\sqrt{1+4a^2}+3}=2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
大润发超市用6800元购进A、B两种计算器各40、80只，这两种计算器的进价、标价如下表。A型：进价(元/只)：30，标价(元/只)：50；B型：进价(元/只)：70，标价(元/只)：100；元旦活动期间，超市决定将A型计算器按标价的9折（标价的90%）出售，为保证这批计算器全部售出后盈利不低于1400元，则B型计算器最多按照标价的多少出售？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.8}
A line with slope $2$ passes through the focus $F$ of the parabola $y^2 = 2px$ $(p > 0)$ and intersects the parabola at points $A$ and $B$. The projections of $A$ and $B$ on the $y$-axis are $D$ and $C$ respectively. If the area of trapezoid $\triangle BCD$ is $6\sqrt{5}$, then calculate the value of $p$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{2}}
A finite sequence of numbers satisfies the following conditions: The sum of any three consecutive terms is negative, and the sum of any four consecutive terms is positive. What is the maximum number of terms in this sequence?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Natural numbers $1, 2, 3, \cdots$ are written consecutively to form a number $123456789101112 \cdots$. At what point does the formed number first become divisible by 72? This number is $\qquad$ _. $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Find the determinant of the matrix representing the linear transformation $x \mapsto \beta x$ in the vector space $\mathbb{Q}(\sqrt{d})$ with basis $\{1,\sqrt{d}\}$, where $\beta$ is an element of $\mathbb{Q}(\sqrt{d})$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a^2 - b^2 d}
If $f:\mathbb{R}\rightarrow \mathbb{R}$ and $f(x)=x+\cos x$, find$$\int^{\pi}_{0}f^{-1}(x) \,\d x.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\dfrac{\pi^2}{2}\)}
Charles Dickens' 200th anniversary was celebrated on February 7, 2012, which was a Tuesday. A year is a leap year if it is divisible by 400 or divisible by 4 but not by 100. Using this rule, calculate the day of the week on which Charles Dickens was born. Express your answer as the full day of the week.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Friday}
Consider a sphere inscribed in a right cone with the base radius of 10 cm and height of 40 cm. The radius of the inscribed sphere can be expressed as $b\sqrt{d} - b$ cm. Determine the value of $b+d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19.5}
During summer break, 610769 kids from Lawrence county go to camp and the other 590796 kids stay home. An additional 22 kids from outside the county attended the camp. About how many kids are in Lawrence county?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1,201,565}
What condition must be imposed on a measure space $(X, \Sigma, \mu)$ to ensure that every function $f \in L_\infty$ satisfies $\sup_{p\ge 1} ||f||_p < \infty$, where $||f||_p = \left(\int_X |f|^p d\mu \right)^{\frac{1}{p}}$ for all $p \ge 1$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mu(X) < \infty}
Given that $\angle BAC = 80^\circ$, $|AB| = |AC|$, $K \in [AB]$, $L \in [AB]$, $|AB|^2 = |AK| \cdot |AL|$, and $|BL| = |BC|$, find $\angle KCB$. If the answer is $x^\circ$, what is the value of $x$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Euhan and Minjune are playing tennis. The first one to reach $25$ points wins. Every point ends with Euhan calling the ball in or out. If the ball is called in, Minjune receives a point. If the ball is called out, Euhan receives a point. Euhan always makes the right call when the ball is out. However, he has a $\frac34$ chance of making the right call when the ball is in, meaning that he has a $\frac14$ chance of calling a ball out when it is in. The probability that the ball is in is equal to the probability that the ball is out. If Euhan won, determine the expected number of wrong callsmade by Euhan.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find the greatest integer \( k \) such that \( \left\lfloor \frac{n}{\sqrt{3}} \right\rfloor + 1 > \frac{n^2}{\sqrt{3n^2 - k}} \) holds for all positive integers \( n \geq 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Alex has 2 cakes - the first cake is cut into 8 slices and the second cake is cut into 12 slices. Alex gives away a fourth of the slices from the first cake and a third of the slices from the second cake to his friends. After that, he gives half of the remaining slices from the first cake and a sixth of the remaining slices from the second cake to his family. Alex decides to store a quarter of the remaining slices of each cake in the freezer for later. He then eats 2 slices from the first cake and 1 slice from the second cake. How many slices of the cake are left including the slices in the freezer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
3. (CZS) A tetrahedron $A B C D$ is given. The lengths of the edges $A B$ and $C D$ are $a$ and $b$, respectively, the distance between the lines $A B$ and $C D$ is $d$, and the angle between them is equal to $\omega$. The tetrahedron is divided into two parts by the plane $\pi$ parallel to the lines $A B$ and $C D$. Calculate the ratio of the volumes of the parts if the ratio between the distances of the plane $\pi$ from $A B$ and $C D$ is equal to $k$.  Second Day
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{k^{3}+3 k^{2}}{3 k+1}}
What are the number of ways in which we can place $7$ apples in $5$ containers such that each container contains at least $1$ apple, given neither apples nor containers are identical?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16800}
If Jack walked 9 miles in 1 hour and 15 minutes with an elevation gain of 600 feet and a descent of 400 feet, what was his adjusted pace in miles per hour, considering that on average, an additional 2 minutes should be added for every 100 feet of elevation gain and 1 minute should be subtracted for every 200 feet of elevation loss?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6.35}
Let $f(x)$ be an odd function that is differentiable on $\mathbb{R}$, and $f'(x)$ is the derivative of $f(x)$. It is known that for $x > 0$, $f(x) < f'(x)$, and $f(1) = e$. The solution set of the inequality $0 < f(\ln(x+ \sqrt{1+x^{2}})) \leqslant e^{\ln(x+ \sqrt{1+x^{2}})}$ is denoted as $M$. Find the number of zeros of $g(x) = \sin 6x$ in the set $M$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
How many ways are there to color the $8$ regions of a three-set Venn Diagram with $3$ colors such that each color is used at least once? Two colorings are considered the same if one can be reached from the other by rotation and/or reflection.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1248}
For each nonnegative integer $n$, we define $A_n = 2^{3n} + 3^{6n+2} + 5^{6n+2}$. Find the greatest common divisor of the numbers $A_0, A_1, \ldots, A_{1999}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Consider the model \( Y_i = \beta + U_i \) for \( i = 1, 2, 3 \), where \( E(U_1) = E(U_2) = E(U_3) = 0 \) and \( \text{Var}(U_1) = 1, \text{Var}(U_2) = 2, \text{Var}(U_3) = 4 \). Determine the best linear unbiased estimate of the variance of \( \beta \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{7}}
A quadrilateral \(A B C D\) is inscribed in a circle with radius 6 and center at point \(O\). Its diagonals \(A C\) and \(B D\) are mutually perpendicular and intersect at point \(K\). Points \(E\) and \(F\) are the midpoints of \(A C\) and \(B D\), respectively. The segment \(O K\) is equal to 5, and the area of the quadrilateral \(O E K F\) is 12. Find the area of the quadrilateral \(A B C D\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
已知$a>0,b>0,c>1$，且$a+b=1$，则$\left( \frac{2a+b}{ab}-3 \right)\cdot c+\frac{\sqrt{2}}{c-1}$的最小值为___________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$4+2\sqrt{2}$}
3. In triangle $A B C$, points $D$ and $E$ lie on the interior of segments $A B$ and $A C$, respectively, such that $A D=1, D B=2, B C=4, C E=2$ and $E A=3$. Let $D E$ intersect $B C$ at $F$. Determine the length of $C F$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Several young men and women gathered around a round table. It is known that exactly 7 women have other women sitting to their left, and 12 women have young men sitting to their left. It is also known that for 75% of the young men, there are women sitting to their right. How many people are sitting at the table?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
Given a connected simple graph $G$, it is known that the number of its edges is $e$. Some chess pieces are placed on each vertex of $G$, and a chess piece can only be placed on one of the vertices of $G$. The only operation you can perform is that if the number of chess pieces on a vertex $v$ of $G$ is not less than the number of its adjacent vertices $d$, then you can choose $d$ chess pieces from $v$ and distribute them to the adjacent vertices, with each vertex receiving one piece. If the number of chess pieces on each vertex of $G$ is less than the number of its adjacent vertices, then you cannot perform any operation.

Find the minimum value of $m$ such that there exists an initial way to place the chess pieces, with a total of $m$ pieces, and you can start from this placement and perform the operation infinitely many times.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e}
Find the minimum size of a subgroup \( H \leq S_{10} \) that contains at least one 9-cycle, one 2-cycle, and acts transitively on the set \( \{1,2,...,10\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3628800}
Let $a, b \in \mathbb{R}$, $c \in [0, 2\pi)$. Find the number of sets of $a, b, c$ that satisfy the equation $2\sin(3x- \frac{\pi}{3}) = a\sin(bx+c)$ for any real number $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $x=-2016$. What is the value of $\left| \ \bigl \lvert { \ \lvert x\rvert -x }\bigr\rvert -|x|{\frac{}{}}^{}_{}\right|-x$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4032
}
What is the remainder when the sum of factorials from $1!$ to $45!$ is divided by $47$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Evaluate the Taylor series of \(f(t)=\exp(t A)\, B \exp(-t A)\) up to the second-order term, given that \(f(0) = B\), \(f'(0) = [A,B]\), and \([A,[A,B]]=0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{B + t [A, B]}
Evaluate the limit: $$\lim \limits_{x \to 0} \int_0^1 \frac{\cos(xt)}{t^2}\,dt$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Evaluate the line integral \( \int_C x \, dy - y \, dx \), where \( C \) is the curve composed of a straight line segment from \((-2, 0)\) to \((0, 0)\), a straight line segment from \((0, 0)\) to \((0, -2)\), and the part of the circle of radius 2, centered at the origin, traversed counterclockwise starting from \((0,-2)\) and ending at \((-2, 0)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6\pi}
How many infinite arithmetic sequences of natural numbers are there in which the ratio of the sum of the first $2n$ terms to the sum of the first $n$ terms does not depend on the choice of $n$, and one of the terms in the sequence is the number 1971?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Two congruent right circular cones each with base radius  $3$  and height  $8$  have axes of symmetry that intersect at right angles at a point in the interior of the cones a distance  $3$  from the base of each cone. A sphere with radius  $r$  lies inside both cones. The maximum possible value for  $r^2$  is  $\frac mn$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{298}
In a spelling contest held in her school, Drew got 20 questions correct, winning her the competition. She got some questions wrong, and her competitor Carla got 14 questions correct, and twice as many questions wrong as the number of questions Drew got wrong. There were 52 questions asked in the competition. How many questions did Drew get wrong?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
A spinner is divided into 4 sectors with areas corresponding to probabilities of 10%, 20%, 30%, and 40%. Calculate the expected number of spins required for the spinner to land on each sector at least once.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{445}{36}}
Divide the set of numbers \(\{1, 2, \cdots, 10\}\) into two groups such that the product of the numbers in the first group, \(p_{1}\), is divisible by the product of the numbers in the second group, \(p_{2}\). Find the minimum value of \(\frac{p_{1}}{p_{2}}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Determine the largest three-digit prime number that is a factor of \( \binom{2000}{1000} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{661}
Let the lengths of the sides of $\triangle A_nB_nC_n$ be $a_n$, $b_n$, $c_n$, for $n=1,2,3,\ldots$. If $b_1 > c_1$, $b_1+c_1=2a_1$, $a_{n+1}=a_n$, $b_{n+1}=\frac{c_n+a_n}{2}$, $c_{n+1}=\frac{b_n+a_n}{2}$, then the maximum value of $\angle A_n$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{3}}
Penelope the pig eats 20 pounds of food per day, which is 10 times more than Greta the goose eats per day. Milton the mouse eats 1/100 as much as Greta the goose eats per day, but Elmer the elephant eats 4000 times more than Milton the mouse does per day. Rosie the rhino eats 3 times more than Greta and twice less than Elmer. On the other hand, Carl the cow eats half as much as Penelope and 5 times more than Greta. If the total amount of food consumed by all animals cannot exceed 1000 pounds per day, and Greta eats x pounds of food per day, what percentage of the total food consumption, rounded to the nearest whole percentage, does Elmer the elephant eat per day, compared to all the other animals combined?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{68\%}
求所有使不等式成立的λ：

$2( x ^ { 3 } + y ^ { 3 } + z ^ { 3 } ) + 3 ( 1 + 3 \lambda )xyz≥(1+λ)(x+y+z)(yz+zx+xy)$这里x,y,z是非负数.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{λ≤1}
Determine the discriminant of the cubic number field $M=\mathbb{Q}(20^{1/3})$, using an integral basis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-10800}
Given the cost function \( C(x) = 125 \sqrt{x^2 + 5^2} + 100|k-x| \), where the cost of underwater cable is \( \alpha \) per kilometer and the cost on land is \( \beta \) per kilometer with \( \alpha > \beta \), determine the largest value of \( k > 0 \) such that the most efficient path is the hypotenuse \( \overline{AF} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{20}{3}}
Determine the values of $b$ for which the set \( F_b = \left\{ \begin{pmatrix} x & by \\ y & x \end{pmatrix} \mid (x,y)\in \mathbb{R}^2 \right\} \) forms a field under matrix addition and multiplication.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{b < 0}
How many polynomials of degree exactly 5 with real coefficients send the set \{1,2,3,4,5,6\} to a permutation of itself?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{714}
A person has 120 light bulbs, each with a lifetime that follows an independent exponential distribution with a mean of 6 hours. When a bulb fails, it is immediately replaced by a new one, with the replacement time being uniformly distributed over (0, 0.25) hours. Approximate the probability that all bulbs have failed by time 750 hours.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
If \[N=\frac{\sqrt{\sqrt{5}+2}+\sqrt{\sqrt{5}-2}}{\sqrt{\sqrt{5}+1}}-\sqrt{3-2\sqrt{2}},\] then $N$ equals
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$1$}
In a game where the winner is the first player to have 7 or more points and is 2 points ahead of the other player, you currently lead with a score of 6-5. Assuming both players have equal skill and each point is equally likely to be won by either player, what is the probability that you will eventually win the game?
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Answer: \boxed{\dfrac{3}{4}}
Consider the category \(\mathbf{FSet}\) of finite sets as an algebraic theory. What is the category of models for this theory?
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Answer: \boxed{\mathbf{Set}}
How many powers of 2 have only the digits 0, 1, 2, 4, and 8 in their decimal representation?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Determine \( \lim_{t \to \infty} u(1,t) \), where \( u(x,t) \) is the solution to the heat equation \( u_t = u_{xx} \) with the initial condition \( u(x,0)=\frac{e^{2x}-1}{e^{2x}+1} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In each cell of  $5 \times 5$  table there is one number from  $1$  to  $5$  such that every number occurs exactly once in every row and in every column. Number in one column is *good positioned* if following holds:
 - In every row, every number which is left from *good positoned* number is smaller than him, and every number which is right to him is greater than him, or vice versa.
 -  In every column, every number which is above from *good positoned* number is smaller than him, and every number which is below to him is greater than him, or vice versa.
What is maximal number of good positioned numbers that can occur in this table?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Evaluate the expression: $$ \sum_{n=-\infty}^\infty \int_{-\pi}^\pi e^{-x^2} \cos(nx) \, dx. $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Twelve couples are seated around a circular table such that all of men are seated side by side, and every women are seated to opposite of her husband. In every step, a woman and a man next to her are swapping. What is the least possible number of swapping until all couples are seated side by side?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
66
}
Evaluate the limit \( \lim_{q\rightarrow 1^-}\frac{\phi^5(q)_{\infty}}{\phi(q^5)_{\infty}} \), where the Euler function is defined as \( \phi(q)=\prod_{n=1}^{\infty}(1-q^{n}) \) and \( q=\exp(2\pi i \tau) \) with \( |q|<1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let  $P$  be a point outside a circle  $\Gamma$  centered at point  $O$ , and let  $PA$  and  $PB$  be tangent lines to circle  $\Gamma$ . Let segment  $PO$  intersect circle  $\Gamma$  at  $C$ . A tangent to circle  $\Gamma$  through  $C$  intersects  $PA$  and  $PB$  at points  $E$  and  $F$ , respectively. Given that  $EF=8$  and  $\angle{APB}=60^\circ$ , compute the area of  $\triangle{AOC}$ .

*2020 CCA Math Bonanza Individual Round #6*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12 \sqrt{3}}
Calculate the integral of \( f(z) = \frac{z^2}{e^{\frac{1}{(z+i)^2}}} \) over the contour \(|z|=2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4\pi}
How many quadratic polynomials with real coefficients are there such that the set of roots equals the set of coefficients? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the dimension of the vector space of functions \( f: \mathbb{R} \to \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\mathfrak{c}}}
Evaluate the limit \(\lim_{x \to {0_+}} \frac{\ln\Bigl(1+ \cfrac{\ln (x \ln a)}{\ln x}\Bigr)}{\ln a}-\frac{\ln\Bigl(1-\cfrac{\ln (x \ln a)}{\ln x}\Bigr)}{\ln a}\) where \(a > 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Let $N$ be the positive integer with 1998 decimal digits, all of them 1; that is, \[ N = 1111\cdots 11. \] Find the thousandth digit after the decimal point of $\sqrt{N}$.
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Answer: \boxed{1}
若 $z \in \mathbf{C}, \arg \left(z^{2}-4\right)=\frac{5 \pi}{6}, \arg \left(z^{2}+4\right)=\frac{\pi}{3}$, 请提供 $z$ 的两个解中实部和虚部之和。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine which function grows faster as $n$ approaches infinity: $f(n) = n!$ or $g(n) = n^{\log_2 n}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n!}
Find the derivative of the function \( f(x) = \arctan\left(\frac{\sqrt{1+x^2}-\sqrt{1-x^2}}{\sqrt{1+x^2}+\sqrt{1-x^2}}\right) \) with respect to \( \arccos(x^2) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}
Let $\alpha$ and $\beta$ be nonzero real numbers such that $2 (\cos \alpha-\cos\beta)+\cos\alpha\cos\beta=1$. Find the value of $\tan \left(\frac{\alpha+\beta}{2}\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}