inclusionAI/Ring-lite-rl-data huggingface dataset:
If circle $$C_{1}: x^{2}+y^{2}+ax=0$$ and circle $$C_{2}: x^{2}+y^{2}+2ax+ytanθ=0$$ are both symmetric about the line $2x-y-1=0$, then $sinθcosθ=$ \_\_\_\_\_\_ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{2}{5}}
$已知函数f(x) = x^3 - 3x^2 + ax + 4, a \in  R.$
$若f(x)>0对x\in (1，2)恒成立，求a的取值范围.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0,+\infty )}
Place as many points as possible on a plane such that any three points do not lie on a single straight line and form the vertices of an isosceles triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given a cube \( ABCD A_1 B_1 C_1 D_1 \). A point is called "equidistant" if there are two vertices of the cube for which this point is the midpoint of the segment connecting them. How many "equidistant" points are there in the cube?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
A triangle  $\vartriangle A_0A_1A_2$  in the plane has sidelengths  $A_0A_1 = 7$ , $A_1A_2 = 8$ , $A_2A_0 = 9$ . For  $i \ge 0$ , given  $\vartriangle A_iA_{i+1}A_{i+2}$ , let  $A_{i+3}$  be the midpoint of  $A_iA_{i+1}$  and let Gi be the centroid of  $\vartriangle A_iA_{i+1}A_{i+2}$ . Let point  $G$  be the limit of the sequence of points  $\{G_i\}^{\infty}_{i=0}$ . If the distance between  $G$  and  $G_0$  can be written as  $\frac{a\sqrt{b}}{c}$  , where  $a, b, c$  are positive integers such that  $a$  and  $c$  are relatively prime and  $b$  is not divisible by the square of any prime, find  $a^2 + b^2 + c^2$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{422}
Andrey, Boris, Vasily, Gennady, and Dmitry played table tennis in pairs so that each pair played against every other pair exactly once. There are no ties in tennis. It is known that Andrey lost exactly 12 times, and Boris lost exactly 6 times. How many times did Gennady win?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given circle $O$: $x^{2}+y^{2}=3$, line $l$: $x+3y-6=0$, and point $P(x_{0},y_{0})∈l$. If there exists a point $Q$ on circle $O$ such that $∠OPQ=60^{\circ}$, determine the range of $x_{0}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0 \leqslant x_0 \leqslant \frac{6}{5}}
In the triangle \(A B C\), angle \(C\) is a right angle, and \(AC: AB = 3: 5\). A circle with its center on the extension of leg \(AC\) beyond point \(C\) is tangent to the extension of hypotenuse \(AB\) beyond point \(B\) and intersects leg \(BC\) at point \(P\), with \(BP: PC = 1: 4\). Find the ratio of the radius of the circle to leg \(BC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{37}{15}}
Let $(X_k)$ be a sequence of random variables as following : $P(X_i=-1)=q$ and $P(X_i=1)=1-q$. I want to calculate $$E[\inf\{n:X_1+X_2+...+X_n>0\}]$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{1-2q}}
Given the shadow length $l$ is equal to the product of the table height $h$ and the tangent value of the solar zenith angle $\theta$, and $\tan(\alpha-\beta)=\frac{1}{3}$, if the shadow length in the first measurement is three times the table height, determine the shadow length in the second measurement as a multiple of the table height.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{3}}
Each square of a chessboard has a knight on it. What is the minimum number of knights that need to be removed so that there isn't a single knight left on the board that can attack exactly three other knights? (A knight attacks squares that are one square away horizontally and two squares vertically, or vice versa.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
已知勾函数$y=x+\frac{{{a}^{2}}}{x}(a>0)$在$(-\infty ,-a)$和$(a,+\infty )$内均为增函数，在$(-a,0)$和$(0,a)$ 内均为减函数。若勾函数$f(x)=x+\frac{t}{x}(t>0)$在整数集合$Z$内为增函数，则实数$t$的取值范围为___________。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$(0,2)$}
A square board is divided by seven lines parallel to one side of the board and seven lines parallel to the other side, into 64 rectangular cells, which are painted in white and black in a checkerboard pattern. The distances between neighboring lines are not necessarily equal, so the cells can be of different sizes. It is known, however, that the ratio of the area of any white cell to the area of any black cell is no more than 2. Find the greatest possible ratio of the total area of the white cells to the total area of the black cells.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{4}}
Given a triangular pyramid $P-ABC$ with its three side edges mutually perpendicular, and $AB=2, AC= \sqrt{7}, BC= \sqrt{13}$, then the volume of the circumscribed sphere of this pyramid is _______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\sqrt{3}\pi}
A ranch has both horses and ponies. Some fraction of the ponies have horseshoes, and exactly 5/8 of the ponies with horseshoes are from Iceland. There are 4 more horses than ponies on the ranch. The minimum possible combined number of horses and ponies on the ranch is 164. What fraction of the ponies have horseshoes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/10}
Once, in a physics and mathematics school, September was declared the month of information technology. Therefore, every Monday in September there was 1 computer science lesson in each class, every Tuesday - 2 lessons, every Wednesday - 3 lessons, every Thursday - 4 lessons, and every Friday - 5 computer science lessons. Saturdays and Sundays were days off at this school. On September 1, there were ceremonial assemblies at the school, so there were no lessons. It is known that there were a total of 64 computer science lessons in each class throughout September. On which day of the week did September 1 fall that year? September has a total of 30 days.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Wednesday}
Lisa bought a new iPhone for $1000. In addition to the iPhone, she bought a phone contract that costs $200/month, a case that costs 20% of the cost of the phone, headphones that cost half as much as the case, a portable charger that costs $60, and an extended warranty plan that costs $150 for the first two years. She also received a 10% discount on the case and headphones due to a promotion. If the phone contract remains the same price, how much will Lisa spend on her phone and related expenses after three years?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$8680}
Find the smallest three-digit number \(n\) such that if the three digits are \(a\), \(b\), and \(c\), then
\[ n = a + b + c + ab + bc + ac + abc. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{199}
We have eight light bulbs, placed on the eight lattice points (points with integer coordinates) in space that are $\sqrt{3}$ units away from the origin. Each light bulb can either be turned on or off. These lightbulbs are unstable, however. If two light bulbs that are at most 2 units apart are both on simultaneously, they both explode. Given that no explosions take place, how many possible configurations of on/off light bulbs exist?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
A and B began riding bicycles from point A to point C, passing through point B on the way. After a while, A asked B, "How many kilometers have we ridden?" B responded, "We have ridden a distance equivalent to one-third of the distance from here to point B." After riding another 10 kilometers, A asked again, "How many kilometers do we have left to ride to reach point C?" B answered, "We have a distance left to ride equivalent to one-third of the distance from here to point B." What is the distance between point A and point C? (Answer should be in fraction form.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{40}{3}"]}
Let \( a_{1}, a_{2}, \ldots, a_{2000} \) be real numbers in the interval \([0,1]\). Find the maximum possible value of
\[ 
\sum_{1 \leq i < j \leq 2000}(j - i) \left| a_{j} - a_{i} \right| 
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1000000000}
In the plane, a rhombus \(ABCD\) and two circles are constructed, circumscribed around triangles \(BCD\) and \(ABD\). Ray \(BA\) intersects the first circle at point \(P\) (different from \(B\)), and ray \(PD\) intersects the second circle at point \(Q\). It is known that \(PD = 1\) and \(DQ = 2 + \sqrt{3}\). Find the area of the rhombus.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
If \(\frac{\left(\frac{a}{c}+\frac{a}{b}+1\right)}{\left(\frac{b}{a}+\frac{b}{c}+1\right)}=11\), and \(a, b\), and \(c\) are positive integers, then the number of ordered triples \((a, b, c)\), such that \(a+2b+c \leq 40\), is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
8 个女孩和 25 个男孩围成一圈,任意两个女孩之间至少站两个男孩, 那么共有多少种不同的排列方法 (只要把圆旋转一下就重合的排法认定是相同的).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{不同的排法总数为 $\frac{25!16!}{9!}$.}
Let $w_1$ and $w_2$ denote the circles $x^2+y^2+10x-24y-87=0$ and $x^2 +y^2-10x-24y+153=0,$ respectively. Let $m$ be the smallest positive value of $a$ for which the line $y=ax$ contains the center of a circle that is externally tangent to $w_2$ and internally tangent to $w_1.$ Given that $m^2=\frac{p}{q},$ where $p$ and $q$ are relatively prime integers, find $p+q.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{169}
Let  $ABC$  be a triangle with  $\angle ABC=90^{\circ}$ . The square  $BDEF$  is inscribed in  $\triangle ABC$ , such that  $D,E,F$  are in the sides  $AB,CA,BC$  respectively. The inradius of  $\triangle EFC$  and  $\triangle EDA$  are  $c$  and  $b$ , respectively. Four circles  $\omega_1,\omega_2,\omega_3,\omega_4$  are drawn inside the square  $BDEF$ , such that the radius of  $\omega_1$  and  $\omega_3$  are both equal to  $b$  and the radius of  $\omega_2$  and  $\omega_4$  are both equal to  $a$ . The circle  $\omega_1$  is tangent to  $ED$ , the circle  $\omega_3$  is tangent to  $BF$ ,  $\omega_2$  is tangent to  $EF$  and  $\omega_4$  is tangent to  $BD$ , each one of these circles are tangent to the two closest circles and the circles  $\omega_1$  and  $\omega_3$  are tangents. Determine the ratio  $\frac{c}{a}$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
The interval in which the function $f(x) = \left(\frac{1}{2}\right)^{x^2 - 2x}$ is monotonically decreasing is $\_\_\_\_\_\_\_\_\_\_.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[1, +\infty)}
Kim uses five different electronic devices in her home – a TV, a refrigerator, an air conditioner, a washing machine, and a laptop. The TV uses 125 watts of electricity per hour and runs for 4 hours a day. The refrigerator uses 100 watts per hour and runs continuously throughout the day. The air conditioner consumes 2000 watts per hour and is used for 6 hours a day. The washing machine uses 500 watts per hour and is operated for 2 hours a day, but only three times a week. Lastly, the laptop uses 60 watts per hour and is in use for 5 hours a day. 

The local electricity provider has implemented a tiered tariff system as follows:
- First 100 kWh/month: 14 cents per kWh
- Next 200 kWh/month: 18 cents per kWh
- Any usage above 300 kWh/month: 22 cents per kWh

Calculate the total cost of electricity for running all these devices for a week, considering the increasing electricity tariff.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$20.07}
Given two positive integers $m$ and $n$, we say that $m \mid\mid n$ if $m \mid n$ and $\gcd(m,\, n/m) = 1$. Compute the smallest integer greater than \[ \sum_{d \mid 2016} \sum_{m \mid\mid d} \frac{1}{m}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52}
Suppose two arithmetic sequences $\{a\_n\}$ and $\{b\_n\}$ have the sum of their first n terms as $S\_n$ and $T\_n$, respectively. If $a\_3 = 2b\_3$ and $\frac{S\_n}{T\_n} = \frac{3n - t}{2n + 2}$ for any $n \in \mathbb{N}^*$, find the value of $\frac{a\_9}{b\_6 + b\_8} + \frac{a\_5}{b\_3 + b\_11}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{12}{7}}
Given that an isosceles trapezoid is circumscribed around a circle, find the ratio of the area of the trapezoid to the area of the circle if the distance between the points where the circle touches the non-parallel sides of the trapezoid is related to the radius of the circle as $\sqrt{3}: 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{8\sqrt{3}}{3\pi}}
Equilateral triangles \( ABF \) and \( BCG \) are constructed outside regular pentagon \( ABCDE \). Compute \( \angle FEG \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48^{\circ}}
Given a sequence $\{a_n\}$ where $a_1 = -1$, $a_2 > a_1$, and $|a_{n+1}-a_n| = 2^{n}$ for $n \in \mathbb{N}^*$, if the subsequence $\{a_{2n-1}\}$ is monotonically decreasing and the subsequence $\{a_{2n}\}$ is monotonically increasing, then find the general term of the sequence $\{a_n\}$: $a_n = \_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{(-2)^n - 1}{3}}
设 $E, F, G$ 分别是正四面体 $A B C D$ 的棱 $A B, B C, C D$ 的中点，求二面角 $C-F G-E$ 的大小。如果答案为 $\pi-\operatorname{arccot} x$，其原始答案的形式为\frac{k\sqrt{n}}{m}，请计算k+m+n的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
$在\triangle ABC中，角A,B,C的对边分别为a,b,c，且满足b \sin B+c \sin C=a \sin A+\sqrt{3}b \sin C.$
求角A的值；
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{\pi }{6}$}
Given that positive real numbers $a$ and $b$ satisfy $\frac{8}{{{{({b+1})}^3}}}+\frac{6}{{b+1}}≤{a^3}+3a$, find the minimum value of $2a+3b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\sqrt{3}-3}
Given that the angle between $\overrightarrow{AB}$ and $\overrightarrow{AC}$ is $90^{\circ}$, $|\overrightarrow{AB}|=2$, and $|\overrightarrow{AC}|=λ \overrightarrow{AB}+μ \overrightarrow{AC}$ $(λ,μ∈R)$, with $\overrightarrow{MF}\cdot \overrightarrow{BC}=0$, find the value of $\frac{\lambda }{\mu }$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
Given positive numbers $a$ and $b$ satisfying $\log _{6}(2a+3b)=\log _{3}b+\log _{6}9-1=\log _{2}a+\log _{6}9-\log _{2}3$, find $\lg \left(2a+3b\right)-\lg \left(10a\right)-\lg \left(10b\right)=\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Let $ \phi : L \to L $ be a linear operator with minimal polynomial $ m_\phi (x) = (x-1)x^8 $. What is the minimal polynomial of $ \phi^3 $?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(x-1)x^8}
Allie picked 100 wildflowers. 13 of the flowers were yellow and white, 17 of the flowers were red and yellow, 14 of the flowers were red and white, 16 of the flowers were blue and yellow, 9 of the flowers were blue and white, 8 of the flowers were red, blue, and yellow, and 6 of the flowers were red, white, and blue. How many more flowers contained the color red than contained the color white, considering the overlapping categories?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the fixed point (x, y) through which the graph of the quadratic function y=x^{2}-(2-m)x+m always passes, regardless of the value of the real number m. Express your answer as an ordered pair (x, y).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, 3)}
There are $n$ seats in a bus, and all tickets have been sold to $n$ passengers. The Absent-Minded Scientist enters first and sits in a randomly chosen seat without looking at their ticket. The remaining passengers enter one by one. If a passenger sees that their seat is vacant, they take their seat. If their seat is occupied, they sit in the first available seat. Find the probability that the last passenger to enter will sit in their assigned seat according to their ticket.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
There are 12 items displayed on a shelf, arranged with 4 items on the upper layer and 8 items on the lower layer. Now, if 2 items are to be selected from the lower layer to be moved to the upper layer, and the relative order of the other items remains unchanged, calculate the number of different ways to adjust the items. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{840}
John decides to go back to school to get his PhD. He first takes 1 year of courses to get acclimated back to school life before starting his PhD. After that, he spends 2 years learning the basics of his field. He then starts research and spends 75% more time on research than he did learning the basics. During his research phase, he takes a 6-month sabbatical for some personal development. After completing his research, he spends half as long on writing his dissertation as his acclimation period. However, he also attends a 3-month conference during his dissertation writing phase. How long did it take him to do everything, including the sabbatical and conference?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7.75}
Find the area of quadrilateral ABCD given that $\angle A = \angle D = 120^{\circ}$, $AB = 5$, $BC = 7$, $CD = 3$, and $DA = 4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{47\sqrt{3}}{4}}
Given that  $n$  is a natural number such that the leftmost digits in the decimal representations of  $2^n$  and  $3^n$  are the same, find all possible values of the leftmost digit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ \{1, 2, 3, 4, 5, 6, 7, 8, 9\} }
A permutation  $(a_1, a_2, a_3, \dots, a_{100})$  of  $(1, 2, 3, \dots, 100)$  is chosen at random. Denote by  $p$  the probability that  $a_{2i} > a_{2i - 1}$  for all  $i \in \{1, 2, 3, \dots, 50\}$ . Compute the number of ordered pairs of positive integers  $(a, b)$  satisfying  $\textstyle\frac{1}{a^b} = p$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Climbing a staircase with four steps, calculate the total number of possible ways to do so, considering different step sequences as distinct arrangements. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
5. [5 points] On a plane with a given rectangular Cartesian coordinate system, a square is drawn with vertices at points $(0 ; 0),(0 ; 69),(69 ; 69)$, and ( $69 ; 0)$. Find the number of ways to choose two grid nodes inside this square (not including its boundary) such that at least one of these nodes lies on one of the lines $y=x$ or $y=69-x$, but neither of the chosen nodes lies on any line parallel to any of the coordinate axes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{601460}
A square has eight points highlighted on its sides, dividing each side into three equal parts. How many right triangles can be drawn with their vertices at these points? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Evaluate the following limit:
$$\lim_{x\to\infty}\left(\sqrt{x^{6} +x^{4}\sqrt{x^{6} +1}}-\sqrt{2x^{6}}\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given that $O$ is the origin, and $P$ is a point on the hyperbola $\frac{x^{2}}{a^{2}}-y^{2}=1 (a > 0)$, two lines are drawn through $P$ parallel to the asymptotes, intersecting the asymptotes at points $A$ and $B$ respectively. The area of the parallelogram $OBPA$ is $1$. Find the eccentricity of the hyperbola.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{5}}{2}}
Evaluate the following sum: $$\frac{1}{2^{99}}\sum_{k=1}^{50} \sum_{r=0}^{k-1}\binom{50}{k}\binom{49}{r}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/2}
If \begin{align}\prod_{k=1}^{n}\tan \left(\frac{k\pi}{2n+1}\right) = \prod_{k = 10}^{19}\cos \left(\frac{k\pi}{10}\right).\end{align} find the value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Given that points A and B lie on the graph of y = \frac{1}{x} in the first quadrant, ∠OAB = 90°, and AO = AB, find the area of the isosceles right triangle ∆OAB.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{5}}{2}}
In $\triangle ABC$, if $a= \sqrt {2}$, $b=2$, and $\sin B+\cos B= \sqrt {2}$, then $A=$ ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac {\pi}{6}}
Given that the domain of the function $f(x)$ is $\mathbb{R}$, and $f(0)=2$, if for any $\left\{x\in \mathbb{R},f(x)+{{f}^{{{'}}}}(x) < 1\right\}$, then the solution set of the inequality $${{e}^{x}}f(x) < {{e}^{x}}+1$$ is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0,+\infty)}
Given that the angle between the generating line and the axis of a cone is $\frac{\pi}{3}$, and the length of the generating line is $3$, find the maximum value of the cross-sectional area through the vertex.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9}{2}}
A simple graph $G$ on 2020 vertices has its edges colored red and green. It turns out that any monochromatic cycle has even length. Given this information, what is the maximum number of edges $G$ could have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1530150}
When Xiao Ma Hu was mentally calculating the product of a two-digit number and a whole ten number, he forgot to add a 0 at the end of the product. As a result, the product was less than the correct product by ____ times.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Let \mathcal{V} be the volume enclosed by the graph $x^{2016}+y^{2016}+z^{2}=2016$. Find \mathcal{V} rounded to the nearest multiple of ten.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{360}
Evaluate the product
$$
\prod_{n=3}^{\infty} \frac{\left(n^{3}+3 n\right)^{2}}{n^{6}-64}
$$
The original answer is in \(\frac{m}{n}\) format, please give the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{79}
Adam is playing Minesweeper on a $9\times9$ grid of squares, where exactly $\frac13$ (or 27) of the squares are mines (generated uniformly at random over all such boards). Every time he clicks on a square, it is either a mine, in which case he loses, or it shows a number saying how many of the (up to eight) adjacent squares are mines.

First, he clicks the square directly above the center square, which shows the number $4$. Next, he clicks the square directly below the center square, which shows the number $1$. What is the probability that the center square is a mine?

[i]Proposed by Adam Bertelli[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{88}{379}}