inclusionAI/Ring-lite-rl-data huggingface dataset:
Let  $r_1, r_2, ..., r_{2021}$  be the not necessarily real and not necessarily distinct roots of  $x^{2022} + 2021x = 2022$ . Let  $S_i = r_i^{2021}+2022r_i$  for all  $1 \le i \le 2021$ . Find  $\left|\sum^{2021}_{i=1} S_i \right| = |S_1 +S_2 +...+S_{2021}|$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2021 \cdot 2022}
On a strip of bus tickets numbered from 000000 to 999999, blue paint marks those tickets where the sum of the digits in even positions equals the sum of the digits in odd positions. Determine the largest difference between the numbers of two consecutive blue tickets.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1001}
Evaluate the limit: \[ \lim _{N\rightarrow \infty } \left( \,{\frac {{N}^{3}}{8\sum _{k=0}^{N-1}({k}^{2}+\left\lfloor k/2\right\rfloor)}} \right) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{8}}
Evaluate the integral \( \displaystyle\int_{|z|=2}\frac{e^{1/z^2}}{1-z}dz \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2\pi i}
All the $7$ digit numbers containing each of the digits $1, 2, 3, 4, 5, 6, 7$ exactly once, and not divisible by $5$ are arranged in increasing order. Find the $200^{th}$ number in the list.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1425673}
Find the largest 9 digit integer in which no two digits are the same and the sum of each pair of adjacent digits is prime. That is, the sum of the first two digits is prime, the sum of the second and third digits is prime, the sum of the third and fourth digits is prime, and so on.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{985,674,321}
Factorize the polynomial $x^{27} + 1$ in the field $\mathbb{F}_3[x]$ using Fermat's Little Theorem and the Freshman's Dream.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(x + 1)^{27}}
Consider a machine with two parts, X and Y. The time until part X fails follows an Exponential distribution with rate \( \lambda = 1 \), and the time until part Y fails follows an Exponential distribution with rate \( \lambda = 0.05 \). When part X fails, it is replaced in 1 time unit, during which the machine is not operational and part Y does not age. When part Y fails, the machine is considered dead. Calculate the expected time until the machine is dead.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Identify the quotient group $\mathbb{C}^*/P$, where $\mathbb{C}^*$ is the multiplicative group of non-zero complex numbers and $P$ is the subgroup of positive real numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S^1}
Evaluate the limit: \[ \lim_{n \to \infty} \frac{5^{\frac{n!}{(2n)!}} - 4^{\frac{1}{n!}}}{3^{\frac{n!}{(2n)!}} - 2^{\frac{1}{n!}}} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the values of $x$ for which the series $$\sum_{n=1}^\infty \frac{\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}\right)(\sin(n x))}{n}$$ converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}}
Use the Laurent series of \( e^{1/z} \) on the punctured plane \( \mathbb{C} - \{0\} \) to compute \( \frac{1}{\pi} \int_0^\pi e^{\cos\theta}\cos(\sin\theta-n\theta) \, d\theta \) for \( n = 0, 1, 2, \ldots \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{n!}}
Determine the limit \( \lim_{\varepsilon \searrow 0} \varepsilon \mathbf{E}(|X|^{1-\varepsilon}) \), where \( X \) is a Cauchy random variable with density \( f(x) = \frac{1}{\pi} \frac{1}{1+x^2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{\pi}}
How many non-isomorphic planar graphs (V, E) exist with 6 vertices (|V| = 6) and 10 edges (|E| = 10), given that each vertex has a degree of 3 or greater?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
What is the Lebesgue measure of the set $A$ in [0,1], defined as the set of numbers with continued fraction expansions that admit an infinite subsequence where each term is divisible by its predecessor?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In rectangle ABCD, a circle is inscribed such that it touches the longer sides AD and BC. The diagonal BD intersects the circle at points E and F, creating the chord EF. If BE = 3, EF = 9, and FD = 16, determine the length of side BC.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
Consider the polynomial \( \prod_{n = 1}^{1996}\left(1 + nx^{3^n}\right) = \sum_{n=0}^{m}a_{n}\,x^{k_{n}} \), where the exponents \( k_{n} \) are in ascending order and the coefficients \( a_{n} \) are nonzero. Determine the coefficient \( a_{1996} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{665280}
[ [Ratio of areas of triangles with a common base or common height] Rearrangement of areas Pentagons

Given a convex pentagon $A B C D E$. The area of each of the triangles $A B C, B C D, C D E, D E A, E A B$ is $S$. Find the area of the given pentagon.

#
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{(5+\sqrt{5})S}{2}}
Let the roots of $ax^2+bx+c=0$ be $r$ and $s$. The equation with roots $ar+b$ and $as+b$ is:
$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^2-bx+ac=0}
The inclination angle of the line $x+ \sqrt {3}y+c=0$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5\pi}{6}}
Determine the values of $b \in \mathbb{R}$ for which the symmetric part of the matrix \( \begin{pmatrix} 4 & -1 & b \\ -1 & 2 & -2 \\ 0 & -2 & 1 \end{pmatrix} \) is positive definite.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
A point is randomly selected on the segment \([6, 11]\). Let \( k \) be the selected value. Determine the probability that the roots of the quadratic equation \( (k^{2}-2k-24)x^{2}+(3k-8)x+2=0 \) satisfy the condition \( x_{1} \leq 2x_{2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
Determine the integer value of \( a \) such that the equation \( x^3 - 3x + a = 0 \) has three distinct roots between 0 and 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A particle performs a random walk on the vertices of a cube. At each step, it remains at its current vertex with probability \( \frac{1}{4} \), or moves to one of its neighboring vertices, each with probability \( \frac{1}{4} \). Let vertices \( A \) and \( D \) be diametrically opposite. If the walk starts at vertex \( A \), determine the mean number of steps until the particle visits vertex \( B \), which is one step away from \( A \), for the first time.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{28}{3}}
$已知函数f(x) = \frac{1}{3}x^{3}-x^{2}.$
$若f(x)在区间(-1, m]上的取值范围是 [-\frac{4}{3},0]，求m的取值范围.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[2,3]}
Evaluate the limit $\lim\limits_{n \to \infty} nx_n$ for the sequence defined by $x_2 \in \left(0,\frac{\pi}{2}\right)$ and $x_{n+1} = \left(1-\frac{1}{n}\right)\sin x_n$ for $n \ge 2$. Note that $x_{n+1} \le \sin x_n \le x_n$ and $|x_{n+1}| \le 1$, which implies the convergence of $\{x_n\}$ to $0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Calculate the integral \( \int_D{f^*\omega } \) where \( f:D \rightarrow \mathbb{R}^3 \) is given by \( f(u,v)=(u,v,u^2+v^2+1) \) with \( D=[0,1]\times [0,1] \), and \( \omega=y\,\mathrm{d}y\wedge \mathrm{d}z+xz\,\mathrm{d}x\wedge \mathrm{d}z \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Find the limiting distribution of \( \frac{Y_n - n}{\sqrt{2n}} \) as \( n \to \infty \), where \( Y_n \sim \chi^2(n) \), using moment generating functions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{N(0,1)}
Define a new function $\$N$ such that $\$N = 0.75N + 2$. Calculate $\$(\$(\$30))$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17.28125}
Evaluate \( {\int }_{0}^{\frac{\pi }{4}}\frac{1}{\cos \theta }\sqrt{\frac{1 + \sin \theta }{\cos \theta }}{d\theta } \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$2(\sqrt{1+\sqrt{2}} - 1)$
}
Given a Brownian motion $X_t$ with $X_0 = 0$ and $X_1 > 0$, find the probability that $X_2 < 0$. Express your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
Determine the least integer $k$ for which the following scenario could hold true: In a chess tournament with $24$ players, every pair of players plays at least $2$ and at most $k$ games against each other. At the end of the tournament, it turns out that every player has played a different number of games.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
点集 $\left\{(x, y) \left\lvert\, \lg \left(x^{3}+\frac{1}{3} y^{3}+\frac{1}{9}\right)=\lg x+\lg y\right.\right\}$ 中元素个数为多少？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let \( f = \sum_{i \ge m} a^i t^t \) (for \( m \in \mathbb{Z} \)) be a formal Laurent power series. Consider its formal differential \( df \). What is the coefficient of degree \(-1\) of the differential form \( \frac{df}{f} \)? Provide an explicit expression for this coefficient.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m}
How many ways are there to distribute 6 distinguishable balls into 2 indistinguishable boxes if no box can hold more than 4 balls?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Find the number of valid strings of length \( n \) formed from the set \( \{A, B, C, D, E, F, G\} \) that contain at least one of the letters A, B, C, or D using the Principle of Inclusion-Exclusion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7^n - 3^n}
Compute the winding number of the loop $\alpha(z)=3z^3+5z^2+2: S^1 \rightarrow \mathbb{C} -$ {$0$} about the origin $0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Calculate the remainder when \(27^{123}\) is divided by \(123\). Given that the Carmichael function of \(123\) is \(40\) and \(27^{40} \equiv 27 \pmod{123}\), use this information to find the remainder.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find a natural number $a$ such that for all natural numbers $n$, the expression $n^4 + a$ is not a prime number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
Let X follow a normal distribution N(-1,6^2), and P(-3 ≤ X ≤ -1) = 0.4. Find the probability P(X ≥ 1). Express your answer as a numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.1}
In the triangular pyramid $SABC$, the height $SO$ passes through point $O$ - the center of the circle inscribed in the base $ABC$ of the pyramid. It is known that $\angle SAC = 60^\circ$, $\angle SCA = 45^\circ$, and the ratio of the area of triangle $AOB$ to the area of triangle $ABC$ is $\frac{1}{2 + \sqrt{3}}$. Find the angle $\angle BSC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75^	heta}
One day in October, Nikita looked at the wall clock and saw that four flies had fallen asleep exactly on the numbers 12, 3, 6, and 9.

Find the probability that exactly 20 minutes after Nikita looked at the clock, two of the flies had already been swept away by the minute hand.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
Let $S(x)$ denote the sum of the digits of a positive integer $x$. Find the maximum possible value of $S(x+2019)-S(x)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Given a positive harmonic function $u$ on the unit disc $U$ with $u(0) = 1$, determine the maximum possible value of $u(1/2)$. Provide your answer as a bound or specific value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let \( \triangle ABC \) be a triangle with \( AB = 8 \), \( BC = 15 \), and \( AC = 17 \). Point \( X \) is chosen at random on line segment \( AB \). Point \( Y \) is chosen at random on line segment \( BC \). Point \( Z \) is chosen at random on line segment \( CA \). What is the expected area of triangle \( XYZ \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
In triangle \( \triangle ABC \), the sides opposite to angles \( A \), \( B \), and \( C \) are \( a \), \( b \), and \( c \) respectively. If the angles \( A \), \( B \), and \( C \) form a geometric progression, and \( b^{2} - a^{2} = ac \), then the radian measure of angle \( B \) is equal to ________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2\pi}{7}}
There is a wooden stick 240 cm long. First, starting from the left end, a line is drawn every 7 cm. Then, starting from the right end, a line is drawn every 6 cm. The stick is cut at each marked line. How many of the resulting smaller sticks are 3 cm long?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Find the last five digits of the sum:
$$1^{100} + 2^{100} + 3^{100} + \ldots + 999999^{100}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
The maximum value of the function $y = \sin 2x + \cos 2x$ where $x \in \mathbb{R}$ is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
如果一个10位正整数的各数码互不相同，且它是11111的倍数，则称之为“好数”.那么“好数”共有＿个.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3456}
设 $a 、 b 、 c \in(0,1], \lambda$ 为实数,使得
$$
\frac{\sqrt{3}}{\sqrt{a+b+c}} \geqslant 1+\lambda(1-a)(1-b)(1-c).
$$
求入的最大值.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{91}
If ‘ + ’ means ‘ × ’ , ‘ - ‘ means ‘ + ’ , ‘ × ’ means ‘ ÷ ’ and ‘ ÷ ’ means ‘ - ‘ then find the value of ; 6 – some number + 8 × 3 ÷ 25 = 5. What is that number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
A dishonest shopkeeper professes to sell three types of pulses (A, B, and C) at the cost price, but he uses different false weights for each. For product A, he uses a false weight of 950 gm for a kg; for product B, he uses a false weight of 900 gm for a kg; and for product C, he uses a false weight of 920 gm for a kg. Additionally, the cost prices of the pulses are as follows: product A costs $20 per kg, product B costs $25 per kg, and product C costs $18 per kg. If the shopkeeper sells 10 kg of each product, calculate his overall percentage gain from selling these products.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8.5\%}
1. Use the numbers $2,3, \cdots, 2019$ to form 1009 fractions, with each number appearing only once in either the numerator or the denominator. From these 1009 fractions, select the largest one. Find the minimum value of this number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1010}{2019}}
For a positive integer  $n$ , there is a school with  $2n$  people. For a set  $X$  of students in this school, if any two students in  $X$  know each other, we call  $X$  *well-formed*. If the maximum number of students in a well-formed set is no more than  $n$ , find the maximum number of well-formed set. 

Here, an empty set and a set with one student is regarded as well-formed as well.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3^n}
Given the parabola $E: x^{2}=4y$ and the circle $F: x^{2}+\left(y-1\right)^{2}=1$, a line $l$ passing through point $F$ intersects the above two curves from left to right at points $A$, $C$, $D$, $B$ respectively. Find the minimum value of $|AC|+2|BD|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{2}}
Express the number 1988 as the sum of some positive integers in such a way that the product of these positive integers is maximal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ 2 \cdot 3^{662} }
It is known that a regular \( n \)-gon can be constructed with straightedge and compass if \( n \) is a prime that is 1 plus a power of 2. It is also possible to construct a \( 2n \)-gon whenever an \( n \)-gon is constructible, or a \( p_{1}p_{2} \cdots p_{m} \)-gon where the \( p_{i} \)'s are distinct primes of the above form. What is the largest \( n \) less than 4,300,000,000 such that a regular \( n \)-gon is constructible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{32}}
The sides of triangle ABC measure 6, 8, and 10. A circle with center P and radius 1 rolls inside triangle ABC, always staying tangent to at least one side. When P first returns to its initial position, what is the path length traversed by P?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
In a certain triangle, the sum of the tangents of the angles is equal to 2016. Estimate (with an accuracy of at least 1 degree) the measure of the largest angle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
8. (5 points) In the cells of a $(2 k+1) \times(2 n+1)$ table, where $k \leqslant n$, the numbers 1, 2, and 3 are arranged such that in any $2 \times 2$ square, there are all three different numbers. What is the maximum value that the sum of the numbers in the entire table can take?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9kn+6n+5k+3}
Given the sets $A=\{|a+1|,3,5\}$ and $B=\{2a+1,a^{2}+2a,a^{2}+2a-1\}$, find $A \cup B$ when $A \cap B = \{2,3\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{-5,2,3,5\}}
Given a cyclic quadrilateral \(ABCD\) with diagonals \(AC\) and \(BD\) intersecting at point \(E\), and the following conditions: \(AC = BC\), \(AD = 5\), \(BE = 12\), \(DE = 3\), find \(\angle BCD\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
Three people, A, B, and C, are sawing wooden sticks of the same thickness. They receive sticks of 8 meters, 10 meters, and 6 meters in length respectively, and are required to saw them into 2-meter sections. After completing the task, A, B, and C have sawn 24, 25, and 27 sections respectively. How many more times did the fastest sawyer saw compared to the slowest sawyer?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given a sample: 1, 3, 4, $a$, 7. Its mean is 4, then the standard deviation of this sample is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given the hyperbola equation x^2/16 - y^2/9 = 1 and point P(4, 4), count the number of lines that pass through P and intersect the hyperbola at only one point. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A corner of a unit cube is chopped off such that the cut runs through the three vertices adjacent to the vertex of the chosen corner. What is the height of the cube when the freshly-cut face is placed on a table?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2\sqrt{3}}{3}}
Using $m$ colors to paint the 6 faces of a cube $V$, with each face painted one color, find the number of distinct colorings of the cube $g(m)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{m^6 + 3m^4 + 12m^3 + 8m^2}{24}}
Find all real numbers \(a\) such that there exists a function \(f : \mathbb{R}^{+} \to \mathbb{R}^{+}\) satisfying the equation \(3(f(x))^{2} = 2f(f(x)) + ax^{4}\) for all \(x \in \mathbb{R}^{+}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 1]}
Calculate the volume of the body bounded by the surfaces $az = a^2 - x^2 - y^2$, $z = 0$, and $x^2 + y^2 + ax = 0$ (where $a > 0$).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ \frac{5 \pi a^3 }{32}}
Determine if there exists a bound on the exponent $a$ such that $x^a = x^b$ holds for all $x$ in any ring $R$, independent of the ring $R$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a = b}
Suppose $G$ is a group defined by the presentation $G=\langle a,b\mid a^2=b^3=e\rangle$. Determine the order of the group $|G|$. \(\boxed{?}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Compute the Euler Characteristic of the blowing up of $\mathbb{C}\mathbb{P}^2$ at $n$ points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 + n}
Determine the dimension of the unitary group $U(n)$, where $U(n) = \{A \in GL_n(\mathbb{C}) \mid A^\ast A = I\}$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^2}
An arithmetic sequence $\{ b_{n}\}$ consists of positive integers. If $b_{12}-b_{1}=330$, find the limit:

$$\lim _{n\rightarrow \infty }\dfrac {n(b^{2}_{n}+1)}{\sum\limits^{n}_{k=1}\left( 2+4+6+\ldots +2k\right) }$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2700}
Calculate the residue of the function \( f(z) = z\exp\left(\frac{1}{1-z}\right) \) at \( z = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}
To commemorate Earth Day, the senior class of Fuxing High School visited the Earth Natural Museum. After the visit, a group of 7 students from a class took a group photo. If student A and student B stand together, and student C stands on the edge, the probability that student C is not adjacent to student A or B is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.8}
10. In a square lawn with side length $a$, there are sprinkler devices installed at each of the four corners. The sprinkler devices can rotate to spray water at a $90^{\circ}$ angle, and each sprinkler can rotate from one side of its corner to the other side, with an effective range of $a$. What is the proportion of the lawn that can be covered simultaneously by all four sprinklers to the entire lawn? $\qquad$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{\\pi+3-3\\sqrt{3}}{3}"]}
Determine the largest possible order of any element in the matrix group ${\rm GL}_n(\mathbb{F}_q)$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{q^n - 1}
Given the function f(x) = A sin(ωx + φ) (A > 0, ω > 0, |φ| < $\frac{\pi}{2}$) whose graph has a y-intercept of 1, its first maximum point to the right of the y-axis is (x₀, 2) and its first minimum point is (x₀ + 3π, -2). If x ∈ [0, $\frac{3\pi}{2}$], find the range of f(x).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[1, 2]}