inclusionAI/Ring-lite-rl-data huggingface dataset:
Evaluate the surface integral \( \iint_S \mathbf{F} \cdot \mathbf{n}\ d \sigma \) where \( \mathbf{F} = xy\mathbf{i}+ z^2\mathbf{j}+y\mathbf{k} \) and \( S \) is the exterior side of the cylinder \( y^2 + z^2 =9 \), bounded by \( x=0 \), \( x=1 \), \( y=0 \), and \( z=0 \) in the first octant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{27}{2}}
Given the sets $A=\{-4, 2a-1, a^{2}\}$ and $B=\{a-5, 1-a, 9\}$, and $\{9\}=A\cap B$, find the value of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3}
Let $x_1$, $x_2$, and $x_3$ be the roots of the polynomial $x^3+3x+1$.  There are relatively prime positive integers $m$ and $n$ such that $\tfrac{m}{n}=\tfrac{x_1^2}{(5x_2+1)(5x_3+1)}+\tfrac{x_2^2}{(5x_1+1)(5x_3+1)}+\tfrac{x_3^2}{(5x_1+1)(5x_2+1)}$.  Find $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
In a polygon with 2013 sides labeled as \(A_1, A_2, \ldots, A_{2013}\), each vertex is marked with a number such that the sum of the numbers on any 9 consecutive vertices is 300. Given that \(A_{13}\) is marked with 13 and \(A_{20}\) is marked with 20, find the number marked on \(A_{2013}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
Given the function $f(x) = \sin(\omega x + \varphi)$ ($\omega > 0$, $|\varphi| \leq \frac{\pi}{2}$), if $(-\frac{\pi}{4}, 0)$ is the center of symmetry of the graph of $f(x)$, and $x = \frac{\pi}{4}$ is an extremum point of $f(x)$, and $f(x)$ is monotonic in the interval $(\frac{5\pi}{18}, \frac{2\pi}{5})$, then the maximum value of $\omega$ is \_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Let $\hat{X}\rightarrow X$ be a degree 10 connected covering space where $X$ is a bouquet of 3 circles. Determine the rank of the free group $\pi_{1}(\hat{X})$. What is the rank of this free group?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Among all triangles formed by selecting any 3 vertices from the 8 vertices of a rectangular prism, how many of them are acute-angled triangles?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
An artist who needs to buy only paint and canvas finds that the cost of canvas has decreased by some percent and the cost of paint has decreased by 60 percent. If paint previously accounted for 4 times as much of the expense of painting as did canvas, the total cost for paint and canvas decreased by 55.99999999999999 percent. By what percent did the cost of canvas decrease?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Let $a_1, a_2, \ldots, a_n$ be a permutation of the numbers 1, 2, \ldots, n. Define the sequence number of $a_i$ (for $i=1,2,\ldots,n$) as the count of numbers that are to the left of $a_i$ and smaller than $a_i$. Given that the sequence number of 8 is 2, the sequence number of 7 is 3, and the sequence number of 5 is 3, calculate the number of different permutations. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}
Two trains, each composed of 15 identical cars, were moving towards each other at constant speeds. Exactly 28 seconds after their first cars met, a passenger named Sasha, sitting in the compartment of the third car, passed by a passenger named Valera in the opposite train. Moreover, 32 seconds later, the last cars of these trains completely passed each other. In which car was Valera traveling?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Calculate the measure of the set \(A = ([-1,2] \times [0,3] \times [-2,4]) \cup ([0,2] \times [1,4] \times [-1,4]) \setminus [-1,1]^3\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
The height of a pyramid is 3, and the area of its base is 9. Find the volume of the prism, one base of which is part of the base of the pyramid, and the opposite base is a section of the pyramid created by a plane passing at a distance of 1 from the apex.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.00}
Find the surface area of the cone defined by \(x^2 + y^2 = z^2\) that lies above the \(xy\)-plane and below the cylinder \(y^2 + z^2 = 16\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16\pi}
Given the equation ax + by + 2c = 0, where c!= 0 and ab - c^2 >= 0, find the maximum value of xy. Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given the parabola $y^2=4x$ with focus $F$, and a moving point $P$ on the parabola, and a point $B(3,4)$, find the minimum value of $|PB|+|PF|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{5}}
Find the least ordinal \( \beta > 0 \) such that \( \alpha \beta = \beta \) for a given \( \alpha > 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha^{\omega}}
In trapezoid $ABCD$, $AB < CD$, $AB \perp BC$, $AB \parallel CD$, and the diagonals $AC$, $BD$ are perpendicular at point $P$. There is a point $Q$ on ray $CA$ past $A$ such that $QD \perp DC$. If
\[\frac{QP}{AP} + \frac{AP}{QP} = \left( \frac{51}{14} \right)^4 - 2,\]
then $\frac{BP}{AP} - \frac{AP}{BP}$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m,n$. Compute $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{61}
有长为 $2^{n}(n=0,1, \cdots, 1009)$ 的线段各三条，则由这 3030 条线段能构成不全等的三角形的个数为 $\qquad$ （用数字作答）。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{510555}
Suppose you have a sequence of independent random variables {$X_i, i\geq1$}, such that $$\Bbb P(X_i=i^3 -1)=i^{-3},$$ $$ \Bbb P(X_i=-1)=1-i^{-3}.$$ Find the value of the constant d to which $\frac1n \sum_{i=1}^n X_i$ converges almost surely.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
On a $ 50 \times 50$ board, the centers of several unit squares are colored black. Find the maximum number of centers that can be colored black in such a way that no three black points form a right-angled triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
98
}
The new PUMaC tournament hosts $2020$ students, numbered by the following set of labels $1, 2, \ldots, 2020$. The students are initially divided into $20$ groups of $101$, with each division into groups equally likely. In each group, the contestant with the lowest label wins, and the winners advance to the second round. Out of these $20$ students, we choose the champion uniformly at random. If the expected value of the champion’s number can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime integers, determine $a + b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2123}
Determine the values of $p$ for which the integral \( \int_{0}^{1}\frac{r^{p+1}}{(1-r^{2})^{\frac{p}{2}-\frac{1}{2}}}dr \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-2, 3)}
Given a natural number  $k$ , find the smallest natural number  $C$  such that  $$ \frac C{n+k+1}\binom{2n}{n+k} $$  is an integer for every integer  $n\ge k$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2k+1}
Given two independent random variables X and Y, where the distribution for X is \( f(x) = 2x \) for \( 0 \leq x \leq 1 \) and 0 otherwise, and the distribution for Y is \( f(y) = \frac{3-y}{3} \) for \( 0 \leq y \leq 3 \), determine the probability that \( X + Y \leq 2 \) given that \( X < Y < 3 - X \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{8}{15}}
Starting with the display "1," calculate the fewest number of keystrokes needed to reach "400".
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
What is the Euler characteristic of the hyperboloid of one sheet?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the residue of the function \( \frac{3w-2}{(w-1)^3} \) at its pole of order 3.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given that the sequence $\{a_n\}$ is a geometric sequence, and the sequence $\{b_n\}$ is an arithmetic sequence. If $a_1-a_6-a_{11}=-3\sqrt{3}$ and $b_1+b_6+b_{11}=7\pi$, then the value of $\tan \frac{b_3+b_9}{1-a_4-a_3}$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\sqrt{3}}
A stick of length $10$ is marked with $9$ evenly spaced marks (so each is one unit apart). An ant is placed at every mark and at the endpoints, randomly facing either right or left. Suddenly, all the ants start walking simultaneously at a rate of $ 1$ unit per second. If two ants collide head-on, they immediately reverse direction (assume that turning takes no time). Ants fall off the stick as soon as they walk past the endpoints (so the two on the end don\u2019t fall off immediately unless they are facing outwards). On average, how long (in seconds) will it take until all of the ants fall off of the stick?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
5
}
The whole numbers from 1 to \( 2k \) are split into two equal-sized groups in such a way that any two numbers from the same group share no more than two distinct prime factors. What is the largest possible value of \( k \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
Given point \( A(-2,2) \), point \( B \) is a moving point on the ellipse \( \frac{x^{2}}{25}+\frac{y^{2}}{16}=1 \), and \( F \) is the left focus. Find the coordinates of \( B \) when \( |AB| + \frac{5}{3}|BF| \) is minimized.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(-\frac{5\sqrt{3}}{2}, 2\right)}
Calculate the area between the function \( f(x) = x \cdot e^{x^2 - 1} \) and its inverse.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{e}}
Given that there are intersections between the circle $x^2+y^2=m$ and the circle $x^2+y^2+6x-8y-11=0$, find the range of the real number $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 < m < 121}
At time  $0$ , an ant is at  $(1,0)$  and a spider is at  $(-1,0)$ .  The ant starts walking counterclockwise around the unit circle, and the spider starts creeping to the right along the  $x$ -axis.  It so happens that the ant's horizontal speed is always half the spider's.  What will the shortest distance ever between the ant and the spider be?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{14}}{4}}
Given a right triangle \( \triangle ABC \) with hypotenuse \( BC = 2\pi \). A circle with center \( A \) and radius \( r \) is tangent to \( BC \). Let \( S \) be the remaining area of \( \triangle ABC \) after being covered by the circle. Find the maximum value of \( S \). Express your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Let \( \alpha \) be an element in an algebraic closure of \( GF(64) \) such that \( \alpha^4 = \alpha + 1 \). Determine the value of \( r \in \mathbb{N} \) for which the field extension \( GF(64)(\alpha) \) is isomorphic to \( GF(2^r) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Let $ABC$ be a triangle with $\angle BCA=20.$ Let points $D\in(BC), F\in(AC)$ be such that $CD=DF=FB=BA.$ Find $\angle ADF.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
∠ADF = 10°
}
Evaluate the integral for $N > 0$:
$$ \frac{\exp\left(-\frac N2\right)}{2\pi N} \iint \exp\left(-\left( \frac {x^2-2xy-2Ny+2y^2}{2N}\right)\right) \,dx\,dy $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A secret facility is a rectangle measuring $200 \times 300$ meters. Outside the facility, there is one guard at each of the four corners. An intruder approached the perimeter of the facility from the outside, and all the guards ran towards the intruder using the shortest paths along the external perimeter (the intruder remained stationary). The total distance run by three of the guards to reach the intruder was 850 meters. How many meters did the fourth guard run to reach the intruder?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}
给定正整数 $n(n \geqslant 2)$. 已知 $2 n$个正实数 $a_{1}, a_{2}, \cdots, a_{2 n}$ 满足: $\sum_{k=1}^{n} a_{2 k-1} \sum_{k=1}^{n} a_{2 k}=\prod_{k=1}^{n} a_{2 k-1}+\prod_{k=1}^{n} a_{2 k}$. 求 $S=\sum_{k=1}^{2 n} \frac{a_{k}^{n-1}}{a_{k+1}}$ 的最小值，其中 $a_{2 n+1}=a_{1}$ 。请给出当 n = 2 时的 S 的最小值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Evaluate the sum \( \sum_{k=1}^{\infty} \arctan\left(\frac{1}{k}\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Given that Connie divided a number by $3$ and got $27$, determine the correct number that she should have multiplied by $3$ to get the correct answer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{243}
Find the minimum value of \( \frac{n}{T(n)} \) for \( n \) in the range \([10000, 99999]\), where \( T(n) \) denotes the number of positive integer divisors of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{140}
John bought 9.25 m of cloth at a cost price of $45 per metre. How much did John spend on the cloth?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$416.25}
Given \( f(x) = x^2 + ax + b \) has two real roots in the interval \([0, 1]\), find the range of values for \( a^2 - 2b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, 2]}
Determine the interval of convergence for the power series \( \sum_{n=1}^{\infty}\frac{(-1)^{\lfloor \log_2 n\rfloor}}{n}x^n \), including the behavior at the endpoints.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-1, 1)}
Find the radius of curvature of the plane curve $x^3 + y^3 = 12xy$ at the point $(0, 0)$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Diana works 10 hours per day on Monday, Wednesday, and Friday, and 15 hours per day on Tuesday and Thursday. She also occasionally takes up additional tasks on weekends, earning a varying amount depending on the task. On one particular week, she earns a total of $1800, with an additional $200 earned from supplementary tasks on Saturday. Furthermore, she also receives a performance-based bonus of $150 for the overall week. There is a major holiday during the week, so Diana earns double her usual hourly rate on Wednesday. Considering all these factors, determine Diana's regular hourly rate.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$20.71}
149. Scattered set. In a printing house, a multiplication line of the form $\overline{a b c} \cdot \overline{b c a} \cdot \overline{c a b}^{*}$ was being set; but the set scattered, and the digits of the product got mixed up. As a result, the product was printed as 2342355286. It is known that $a>b>c$ and that the units digit of the printed product is the correct digit 6. Restore the true value of the product.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{328245326}
Given the ellipse $C:\dfrac{x^{2}}{4}+\dfrac{y^{2}}{3}=1$ with its left and right vertices as $A$ and $B$, respectively, and $F$ as the right focus of the ellipse $C$. There is a moving point $P$ on the circle $x^{2}+y^{2}=4$, where $P$ is different from points $A$ and $B$. The line $PA$ intersects the ellipse $C$ at point $Q$ (different from point $A$). If the slope of line $QF$ exists, then the range of $\dfrac{k_{{PB}}}{k_{{QF}}}$ is \_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty,0) \cup (0,1)}
Monsieur and Madame Dubois are traveling from Paris to Deauville, where their children live. Each is driving their own car. They depart together and arrive in Deauville at the same time. However, Monsieur Dubois spent on stops one-third of the time during which his wife continued driving, while Madame Dubois spent on stops one-quarter of the time during which her husband was driving.

What is the ratio of the average speeds of each of their cars?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8/9}
Find a value of $k$ such that the set $S = \{A_1, A_2, A_3\}$, where each $A_i$ is a $2 \times 2$ matrix containing $k$, generates a vector space. Ensure that $k$ appears at least once in each matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the expression $$(x-y)^5 + 5(x-y)^3 - 2(x-y)^2 + 4(x-y)$$ given that $$\sqrt[3]{\frac{x+y}{x-y}} + \sqrt[3]{\frac{x-y}{x+y}} = x+y$$ and $$\sqrt{xy} = 1.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Consider the quadratic equation $x^{2}-(r+7) x+r+87=0$ where $r$ is a real number. This equation has two distinct real solutions $x$ which are both negative exactly when $p<r<q$, for some real numbers $p$ and $q$. What is the value of $p^{2}+q^{2}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8098}
Let $f$ be a function taking the nonnegative integers to the nonnegative integers, such that
\[2f(a^2 + b^2) = [f(a)]^2 + [f(b)]^2\]for all nonnegative integers $a$ and $b.$

Let $n$ be the number of possible values of $f(25),$ and let $s$ be the sum of the possible values of $f(25).$  Find $n \times s.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{153}
What is the smallest number that can be formed by using natural numbers 2 and 4?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
There are 100 towns labeled from 1 to 100. The cost to build a bridge between town "i" and town "j" is \( \max\{|i - j|, 4\} \). However, if a bridge is built between town "i" and town "2i", the cost is 2. What is the minimum cost to connect all the towns in a single connected graph?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{296}
Let $\gamma_1$ and $\gamma_2$ be circles centered at $O$ and $ P$ respectively, and externally tangent to each other at point $Q$. Draw point $D$ on $\gamma_1$ and point $E$ on $\gamma_2$ such that line $DE$ is tangent to both circles. If the length $OQ = 1$ and the area of the quadrilateral $ODEP$ is $520$, then what is the value of length $PQ$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
Sasha discovered that there are exactly $n$ functional buttons with digits left on the calculator. It turns out that any natural number from 1 to 99999999 can either be entered using only the functional buttons, or obtained as the sum of two natural numbers, each of which can be entered using only the functional buttons. What is the smallest $n$ for which this is possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
It takes Bryan some minutes to walk from his house to the bus station. Then he rides the bus for 20 minutes. After that, he walks the same amount of time from the bus station to his job. It takes the same amount of time in the morning and the evening. Bryan spends 365 hours per year traveling to and from work, if he works every day. How many minutes does it take Bryan to walk from his house to the bus station and from the bus station to his job?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Determine the order of the group \( G = \langle a, b : a^3 = b^2 = (ab)^3 = e \rangle \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Calculate the closed loop integral \( \int_\gamma Pdx + Qdy \) for the path \( \gamma: (2, 0) \to (-1,1) \to (0, -2) \to (1, 2) \to (-2, -1) \to (2, 0) \) where \( P(x,y)=  \frac{x-y}{x^2+y^2} \) and \( Q(x,y)=  \frac{x+y}{x^2+y^2} \). Use Green's Theorem to determine the value of the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Xiao Li was doing a subtraction problem and mistook the tens digit 7 for a 9 and the ones digit 3 for an 8, resulting in a difference of 76. The correct difference is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51}
Consider the following flowchart: INPUT $\rightarrow$ Subtract $8 \rightarrow \square \rightarrow$ Divide by $2 \rightarrow \square$ Add $16 \rightarrow$ OUTPUT. If the OUTPUT is 32, what was the INPUT?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
a jar full of whisky contains 40 % alcohol . a part of this whisky is replaced by another containing 19 % alcohol and now the percentage of alcohol was found to be 26 % . the quantity of whisky replaced is :
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2/3}
In a diagram, several circles are connected by line segments. Choose a natural number \( n \) and place different natural numbers in the circles such that:

- If two numbers \( a \) and \( b \) are not connected by a line segment, then \( a - b \) is coprime with \( n \).
- If two numbers \( a \) and \( b \) are connected by a line segment, then \( a - b \) shares a common natural divisor greater than 1 with \( n \).

Determine the smallest natural number \( n \) for which such an arrangement is possible.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
If
\[
\sum_{n=1}^{\infty}\frac{\frac{1}{1} + \frac{1}{2} + \dots + \frac{1}{n}}{\binom{n+100}{100}} = \frac{p}{q}
\]
for relatively prime positive integers $p,q$, find $p+q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9901}
What is the smallest number of plastic snap-together cubes that can be snapped together such that only receptacle holes are showing, with all protruding snaps hidden? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Joel is rolling a 6-sided die. After his first roll, he can choose to re-roll the die up to 2 more times. If he rerolls strategically to maximize the expected value of the final value the die lands on, the expected value of the final value the die lands on can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
$2020$ magicians are divided into groups of $2$ for the Lexington Magic Tournament. After every $5$ days, which is the duration of one match, teams are rearranged so no $2$ people are ever on the same team. If the longest tournament is $n$ days long, what is the value of $n?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
10095
}
Find the difference between the maximum and minimum values of the function \( f(x) = \sqrt{x^2 + 2x + 2} - \sqrt{x^2 - 3x + 3} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}