inclusionAI/Ring-lite-rl-data huggingface dataset:
In a granary, there are 3318 stones of grain. A sample of grain is taken, which contains 168 grains, and 22 of them are imperfect. Using this information, calculate the approximate amount of imperfect grains in the granary. Express your answer in stones, rounded to the nearest whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{434}
Call a set of integers spacy if it contains no more than one out of any three consecutive integers. How many subsets of $\{1,2,3,\cdots,12\},$ including the empty set, are spacy?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{129}
The area of the region bounded by the graph of\[x^2+y^2 = 3|x-y| + 3|x+y|\]is $m+n\pi$, where $m$ and $n$ are integers. What is $m + n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$54$}
The sequence $\{a\_n\}$ satisfies: $a\_1= \frac {1}{2}$, $a\_{n+1}= \frac {1+a\_n}{1-a\_n} (n\geqslant 1)$, find $a\_{2016}=$ \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{- \frac {1}{3}}
From 49 balls numbered 1 to 49, select some balls and arrange them in a circle so that the product of the numbers on any two adjacent balls is less than 100. What is the maximum number of balls that can be selected?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Given the function $f(x)=\cos(\omega x+\varphi)$, where $\omega \in N$, $\varphi \in [0,\pi]$, there are exactly two extreme points in the interval $x \in \left(-\frac{2}{3}\pi,\frac{1}{3}\pi\right)$, and $f\left(-\frac{2}{3}\pi\right)+f\left(\frac{1}{3}\pi\right)=0$. Find the set of all possible values of $\varphi$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left\{0,\frac{5\pi}{6},\pi\right\}}
In measuring the sides of a rectangle, one side is taken 12% in excess and the other 5% in deficit. What is the error percentage in the area calculated from these measurements?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6.4\%}
In triangle \(ABC\), points \(M\), \(N\), and \(P\) are chosen such that \(M\) and \(N\) lie on sides \(AC\) and \(BC\) respectively, and \(P\) lies on segment \(MN\). The following conditions are satisfied:

\[
\frac{|AM|}{|MC|} = \frac{|CN|}{|NB|} = \frac{|MP|}{|PN|}
\]

Find the area of triangle \(ABC\) given that the areas of triangles \(AMP\) and \(BNP\) are \(T\) and \(Q\) respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(T^{1/3} + Q^{1/3})^3}
Write down an irrational number $x$ such that $1 \lt x \lt 4$. $x$ can be ____ (just write down one $x$ that satisfies the condition).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Determine the number of integers $n$ between 1500 and 1515 such that $5$ divides $\left\lfloor\frac{\sqrt{3}}{2}(1+\sqrt{3})^n\right\rfloor$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In the right trapezoid \(ABCD\) (with \(BC\) parallel to \(AD\) and \(AB\) perpendicular to \(AD\)), the shorter base \(AD\) is equal to 3, and the leg \(CD\) is equal to 6. Point \(E\), the midpoint of side \(CD\), is connected by a line segment to point \(B\).

It is known that the angle \(CBE\) is \(\alpha\). Find the area of trapezoid \(ABCD\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72 \sin(\alpha) \cos^3(\alpha)}
Given Liam has written one integer three times and another integer four times. The sum of these seven numbers is 131, and one of the numbers is 17, determine the value of the other number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
We make colored cubes according to the following specifications:

1. Each face of the cube is divided into two triangles by a diagonal, such that every drawn diagonal connects with two other such diagonals at each end point.
2. Each face's two triangles must be of different colors.
3. Triangles that are adjacent along an edge must be of the same color.

How many different cubes can be made using 6 colors? (Two cubes are not considered different if they can be positioned so that their painted faces look the same from any viewpoint.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Find the area bounded by the graph of \( y = \arcsin(\cos x) \) and the \( x \)-axis on the interval \( \left[0, 2\pi\right] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi^2}{2}}
The distance between every two telephone poles by the roadside is 50 meters. Xiao Wang is traveling at a constant speed by car. He counted 41 telephone poles within 2 minutes after seeing the first pole. How many meters per hour is the car traveling?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60000 \, \text{meters per hour}}
Solve the fractional equation application problem. A maglev train is a train that is propelled by magnetic levitation force. The construction of maglev trains is a brilliant leap in the history of China's transportation development. The distance between station $A$ and station $B$ is $30$ km. The average speed of the maglev train running between them is $6.25$ times the average speed of the subway. Moreover, the time taken to ride the maglev train is $\frac{7}{10}$ hours less than the time taken to ride the subway. Find the average speed of the maglev train.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{225 \text{ km/h}}
There are three piles of peaches with the following counts: $93, 70, 63$. A monkey moves peaches among the 3 piles. It is known that the monkey can move at most 5 peaches at a time, and will eat 1 peach while carrying it from one pile to another. When the number of peaches in the 3 piles is equal, how many peaches will the monkey have eaten at least?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In the trapezoid \(ABCD\), the bases \(AD\) and \(BC\) are 8 and 18 respectively. It is known that the circumcircle of triangle \(ABD\) is tangent to the lines \(BC\) and \(CD\). Find the perimeter of the trapezoid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
Haman's father has an egg business supplying the local market. On a Wednesday morning, his father sends him to go and collect 10 trays of eggs for sale from their store. While loading the eggs into the car, he accidentally drops some trays. He calls his father telling him this, and is told to add 7 more trays for sale. They sold 540 eggs that day. How many trays did Haman accidentally drop?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given that the ellipse $\frac{x^{2}}{a} + \frac{y^{2}}{b} = 1 (a > b > 0)$ and the hyperbola $\frac{x^{2}}{m} - \frac{y^{2}}{n} = 1 (m, n > 0)$ share the same foci $F\_1$, $F\_2$, and $P$ is the point of intersection of the two curves, find the value of $|PF\_1| \cdot |PF\_2|$. Express your answer as a single algebraic expression in terms of $a$ and $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a - m}
Find the polar coordinates of the center of the circle given by the equation $\rho = \cos\left(\theta + \frac{\pi}{3}\right)$. Express your answer in polar coordinates as $(\rho, \theta)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(\frac{1}{2}, -\frac{\pi}{3}\right)}
Michael picks a random subset of the complex numbers $\left\{1, \omega, \omega^{2}, \ldots, \omega^{2017}\right\}$ where $\omega$ is a primitive $2018^{\text {th }}$ root of unity and all subsets are equally likely to be chosen. If the sum of the elements in his subset is $S$, what is the expected value of $|S|^{2}$? (The sum of the elements of the empty set is 0.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1009}{2}}
Let the function $f(n)=k$ where $n\in\mathbb{N}^*$ and $k$ is the $n$-th digit after the decimal point of $\pi$, $\pi=3.141\ 592\ 653\ 5\ldots$, then $\{f\ldots f[f(10)]\}=$ ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $\{a_{n}\}$ be an integer sequence such that for any $n \in \mathbf{N}^{*}$, the condition \((n-1) a_{n+1} = (n+1) a_{n} - 2 (n-1)\) holds. Additionally, \(2008 \mid a_{2007}\). Find the smallest positive integer \(n \geqslant 2\) such that \(2008 \mid a_{n}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{501}
Find the integer \( n \), \( -150 < n < 150 \), such that \( \tan n^\circ = \tan 1340^\circ \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-100}
Tommy's mother lets him ride his bike 2 blocks north of his block, 3 blocks east, 2 blocks west, and 2 blocks south. His friend's father lets his friend ride his bike in a certain area. The area his friend can ride in is 4 times greater than Tommy's. How many square blocks can his friend ride his bike in?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Define a sequence recursively by $t_1 = 20$, $t_2 = 21$, and\[t_n = \frac{5t_{n-1}+1}{25t_{n-2}}\]for all $n \ge 3$. Then $t_{2020}$ can be expressed as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{626}
At 8:00 AM, Xiao Cheng and Xiao Chen set off from locations A and B respectively, heading towards each other. They meet on the way at 9:40 AM. Xiao Cheng says: "If I had walked 10 km more per hour, we would have met 10 minutes earlier." Xiao Chen says: "If I had set off half an hour earlier, we would have met 20 minutes earlier." If both of their statements are correct, how far apart are locations A and B? (Answer in kilometers).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}
How many non-singular $3 \times 3$ matrices exist with elements either $1$ or $-1$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{192}
Given \( x, y, z \in \mathbb{Z}_{+} \) with \( x \leq y \leq z \), how many sets of solutions satisfy the equation \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{2}\) ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
6. How many pairs of natural numbers $(a ; b)$ exist such that the number $5 a-3$ is divisible by $b$, and the number $5 b-1$ is divisible by $a$? Enter the number of such pairs of numbers in the provided field.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Given a parabola $x^2=4y$ with focus $F$ and the point $A(-1, 8)$, if $P$ is a point on the parabola, then the minimum value of $|PA| + |PF|$ is \_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
There are a certain number of identical plastic bags that can be nested within each other. If all the other bags are inside one of the bags, we call this situation a "bag of bags." Calculate the number of ways to make a "bag of bags" from 10 bags.

Explanation: Use parentheses to denote a bag.

If we had one bag, the way to make a "bag of bags" would be: (). Two bags can be nested in only one way: (()).

Three bags can be nested in two different ways: $(()())$ and $((()))$, and so on.

The order of bags inside the bag matters. For example, the arrangement $(())())$ is different from $(()(()))$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16796}
A brand of orange juice is available in shop $A$ and shop $B$ at an original price of $\$2.00$ per bottle. Shop $A$ provides a "buy 4 get 1 free" promotion and shop $B$ provides a $15\%$ discount if one buys 4 bottles or more. Find the minimum cost (in cents) if one wants to buy 13 bottles of the orange juice.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2160}
The intersection of two squares with a perimeter of $8$ is a rectangle with a diagonal length of $1$. Given that the distance between the centers of the two squares is $2$, the perimeter of the rectangle can be expressed as $P$. Find $10P$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Find the limit of the following integral as $n$ approaches infinity:

$$ \lim_{n\to\infty} \int_0^1 \dfrac{n x^p+x^q}{x^p+n x^q} dx$$

where $q<p+1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{p-q+1}}
Consider the grid of points $X = \{(m,n) \mid 0 \leq m,n \leq 4 \}$. We say a pair of points $\{(a,b),(c,d)\}$ in $X$ is a knight-move pair if $(c = a \pm 2$ and $d = b \pm 1)$ or $(c = a \pm 1$ and $d = b \pm 2)$. The number of knight-move pairs in $X$ is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Find the gradient of the function $$f(A)=Trace\left[\left(AB\right)^{T}Q\left(AB\right)\right]$$ where $Q = Q^T>0$ with respect to $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2QABB^T}
Javier is excited to visit Disneyland during spring break. He plans on visiting five different attractions, but he is particularly excited about the Space Mountain ride and wants to visit it twice during his tour before lunch. How many different sequences can he arrange his visits to these attractions, considering his double visit to Space Mountain?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{360}
Let $Q$ be a quadratic polynomial. If the sum of the roots of $Q^{100}(x)$ (where $Q^i(x)$ is defined by $Q^1(x)=Q(x)$, $Q^i(x)=Q(Q^{i-1}(x))$ for integers $i \geq 2$) is $8$ and the sum of the roots of $Q$ is $S$, compute $|\log_2(S)|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{96}
Alice and Mary were searching attic and found scale and box with weights. When they sorted weights by mass, they found out there exist  $5$  different groups of weights. Playing with the scale and weights, they discovered that if they put any two weights on the left side of scale, they can find other two weights and put on to the right side of scale so scale is in balance. Find the minimal number of weights in the box
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Given proposition $p$: The equation $x^{2}-ax+4=0$ has real roots; proposition $q$: The function $y=2x^{2}+ax+4$ is increasing on the interval $(3,+\infty)$. If "proposition $p$ or proposition $q$" is true, and "proposition $p$ and proposition $q$" is false, find the range of the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty,-12)\cup(-4,4)}
Consider a modified octahedron with an additional ring of vertices. There are 4 vertices on the top ring, 8 on the middle ring, and 4 on the bottom ring. An ant starts at the highest top vertex and walks down to one of four vertices on the next level down (the middle ring). From there, without returning to the previous vertex, the ant selects one of the 3 adjacent vertices (excluding the one it came from) and continues to the next level. What is the probability that the ant reaches the bottom central vertex?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
It is known that from these five segments it is possible to form four different right triangles. Find the ratio of the largest segment to the smallest.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find the greatest value of the positive integer $x$, such that the number \( A = 2^{182} + 4^x + 8^{700} \) is a perfect square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2008}
Two beads, each of mass  $m$ , are free to slide on a rigid, vertical hoop of mass  $m_h$ . The beads are threaded on the hoop so that they cannot fall off of the hoop. They are released with negligible velocity at the top of the hoop and slide down to the bottom in opposite directions. The hoop remains vertical at all times. What is the maximum value of the ratio  $m/m_h$  such that the hoop always remains in contact with the ground? Neglect friction.
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Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{2}}
Find every real values that  $a$  can assume such that  $$ \begin{cases}
x^3 + y^2 + z^2 = a
x^2 + y^3 + z^2 = a
x^2 + y^2 + z^3 = a
\end{cases} $$  has a solution with  $x, y, z$  distinct real numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(\frac{23}{27}, 1\right)}
Given a biased coin with probabilities of $\frac{3}{4}$ for heads and $\frac{1}{4}$ for tails, determine the difference between the probability of winning Game A, which involves 4 coin tosses and at least three heads, and the probability of winning Game B, which involves 5 coin tosses with the first two tosses and the last two tosses the same.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{89}{256}}
The function $y=2\cdot a^{|x-1|}-1 (a > 0, a \neq 1)$ passes through the fixed point $\boxed{\text{answer}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1,1)}
Given the function $$ F(x) = \int_x^{\sin x} \left( \int_0^{\sin t} \sqrt{1+u^4} \, du \right) dt, $$ find the second derivative $F''(\pi)$ using the fundamental theorem of calculus.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
From the numbers 1, 2, 3,..., 7, select several numbers such that the sum of the even numbers is equal to the sum of the odd numbers. How many combinations satisfy this condition? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
The number $2013$ is expressed in the form   $2013 = \frac {a_1!a_2!...a_m!}{b_1!b_2!...b_n!}$,where $a_1 \ge a_2 \ge \cdots \ge a_m$ and $b_1 \ge b_2 \ge \cdots \ge b_n$ are positive integers and $a_1 + b_1$ is as small as possible. What is $|a_1 - b_1|$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
For each integer $n$ greater than 1, let $F(n)$ be the number of solutions of the equation $\sin x = \sin nx$ on the interval $[0, \pi]$.  What is $\sum_{n=2}^{2007} F(n)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2{,}016{,}532}
If the fractional equation in terms of $x$, $\frac{{x+5}}{{x-3}}=2-\frac{m}{{3-x}}$, has a positive root, then the value of the constant $m$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Consider 9 points in the plane with no alignment. What is the minimum value of $n$ such that if we color $n$ edges connecting two of the points in red or blue, we are sure to have a monochromatic triangle regardless of the coloring?

(IMO 1992)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
What is the largest possible number of subsets of the set \(\{1, 2, \ldots, 2n + 1\}\) such that the intersection of any two subsets consists of one or several consecutive integers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(n+1)^2}
Compute the number of positive integers that divide at least two of the integers in the set $\{1^1, 2^2, 3^3, 4^4, 5^5, 6^6, 7^7, 8^8, 9^9, 10^{10}\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Call a \(2n\)-digit base-10 number special if we can split its digits into two sets of size \(n\) such that the sum of the numbers in the two sets is the same. Let \(p_{n}\) be the probability that a randomly-chosen \(2n\)-digit number is special. (We allow leading zeros in \(2n\)-digit numbers). The sequence \(p_{n}\) converges to a constant \(c\). Find \(c\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Compute the definite integral:

$$
\int_{\pi / 2}^{\pi} 2^{4} \cdot \sin ^{6} x \cos ^{2} x \, dx
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5\pi}{16}}
Find the minimum number of terms required in the Maclaurin series expansion of $f(x) = \log \left(\frac{1+x}{1-x}\right)$ to approximate the value of $\log(2)$ with six exact decimal places when $x = \frac{1}{3}$. Note that $\log$ represents the natural logarithm ($\ln$).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Zhendarov R.G.

Cells of a $9 \times 9$ board are painted in a checkerboard pattern with black and white colors (corner cells are white). What is the minimum number of rooks needed to be placed on this board so that all white cells are under attack by these rooks? (A rook attacks all cells in the row and column where it is placed.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
3. [30] Let $M$ be a $2014 \times 2014$ invertible matrix, and let $\mathcal{F}(M)$ denote the set of matrices whose rows are a permutation of the rows of $M$. Find the number of matrices $F \in \mathcal{F}(M)$ such that $\operatorname{det}(M+F) \neq 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2013!!)^{2}}
Let \(ABCD\) be a convex quadrilateral such that \(AB + BC = 2021\) and \(AD = CD\). We are also given that \(\angle ABC = \angle CDA = 90^\circ\). Determine the length of the diagonal \(BD\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2021}{\sqrt{2}}}
Given that \( \boldsymbol{m} \) is a non-zero vector, \( \boldsymbol{n} \) is a unit vector, \( \boldsymbol{m} \neq \boldsymbol{n} \), and the angle between \( \boldsymbol{m} \) and \( \boldsymbol{m} - \boldsymbol{n} \) is \( 60^{\circ} \), with \( |\boldsymbol{m}| \in (0, a] \), find the minimum value of \( a \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2\sqrt{3}}{3}}
在正方体的每个面上随机填入 1,2,\cdots,6 中的一个数字（各个面上的数字互不相同）. 再对八个顶点进行编号，使得每个顶点的编号数为其相邻三个面上的数字之积. 则八个顶点的编号数之和的最大值为__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{343}
In the isosceles right triangle \(AOB\), points \(P, Q, R,\) and \(S\) are chosen on the sides \(OB, OA,\) and \(AB\) respectively, such that \(PQRS\) forms a square. If the lengths of \(OP\) and \(OQ\) are \(a\) and \(b\) respectively, and the area of the square \(PQRS\) is \(\frac{2}{5}\) of the area of the triangle \(AOB\), determine the value of \(\frac{a}{b}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
People were sitting in a circle. The 7th one is directly opposite to the 18th one. How many people were there in that group?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Given the planar vectors $\overrightarrow {e_{1}}$ and $\overrightarrow {e_{2}}$ that satisfy $|\overrightarrow {e_{1}}| = |3\overrightarrow {e_{1}} + \overrightarrow {e_{2}}| = 2$, determine the maximum value of the projection of $\overrightarrow {e_{1}}$ onto $\overrightarrow {e_{2}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{4\sqrt{2}}{3}}
The power function $f(x) = (2m^2 + m)x^m$ is monotonically increasing on the interval $[0, +\infty)$. Determine the value of $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}