inclusionAI/Ring-lite-rl-data huggingface dataset:
Four circles of radius 1 are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle? Express your answer as a common fraction in simplest radical form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["1 + \\sqrt{2}"]}
Let $S=\\{(a,b)|a=1,2,\\dots,n,b=1,2,3\\}$. A [i]rook tour[\/i] of $S$ is a polygonal path made up of line segments connecting points $p_1,p_2,\\dots,p_{3n}$ is sequence such that \r\n\r\n(i) $p_i\\in S,$ \r\n\r\n(ii) $p_i$ and $p_{i+1}$ are a unit distance apart, for $1\\le i\u003C3n,$ \r\n\r\n(iii) for each $p\\in S$ there is a unique $i$ such that $p_i=p.$ \r\n\r\nHow many rook tours are there that begin at $(1,1)$ and end at $(n,1)?$\r\n\r\n(The official statement includes a picture depicting an example of a rook tour for $n=5.$ This example consists of line segments with vertices at which there is a change of direction at the following points, in order: $(1,1),(2,1),(2,2),(1,2), (1,3),(3,3),(3,1),(4,1), (4,3),(5,3),(5,1).$)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$2^{n-2}$}
Let \( p(x) \) be a polynomial of degree strictly less than 100 such that it does not have \( x^3-x \) as a factor. If \( \frac{d^{100}}{d x^{100}} \frac{p(x)}{x^3-x} =\frac{f(x)}{g(x)} \) for some polynomials \( f(x) \) and \( g(x) \), find the smallest possible degree of \( f(x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
Determine the distribution of the sum of 50 independent and identically distributed uniform random variables on the interval (2, 5).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{N(175, 37.5)}
In the Cartesian coordinate plane $(xOy)$, a moving point $P$ passes through the chords $PA$ and $PB$ of the circles $C_{1}$: $x^{2}+y^{2}+2x+2y+1=0$ and $C_{2}$: $x^{2}+y^{2}-4x-6y+9=0$ respectively, where $A$ and $B$ are the points of intersection. If $|PA|=|PB|$, find the minimum value of $|OP|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{5}}
Let  $n$  be a positive integer. Every square in a  $n \times n$ -square grid is either white or black.
How many such colourings exist, if every  $2 \times 2$ -square consists of exactly two white and two black squares? 
The squares in the grid are identified as e.g. in a chessboard, so in general colourings obtained from each other by rotation are different.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the closed form expression for the derivative of the determinant with respect to matrix $A$:
\[ \frac{\partial \det(X - XA'(AXA')^{-1}AX)}{\partial A} \] where $X$ is an $n \times n$ full rank matrix and $A$ is an $m \times n$ full rank matrix with $m \le n$. Assume $rank(X) = n$ and $rank(A) = m$. Provide your answer in terms of $X$ and $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A certain debt will be paid in 52 installments from January 1 to December 31 of a certain year. Each of the first 8 payments is to be $410; each of the remaining payments is to be a certain amount more than each of the first 8 payments. The average (arithmetic mean) payment that will be made on the debt for the year is $465. What is the difference between the remaining payments and the first 8 payments?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65}
A random sample of size 7 is drawn from a distribution with probability density function \( f_{\theta}(x) = \frac{1+x^2}{3\theta(1+\theta^2)} \) for \( -2\theta \le x \le \theta \) and \( x > 0 \), and zero otherwise. The observations are 12, -54, 26, -2, 24, 17, -39. What is the maximum likelihood estimate of \( \theta \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Let an integer $n > 1$ be given. In the space with orthogonal coordinate system $Oxyz$ we denote by $T$  the set of all points $(x, y, z)$ with $x, y, z$ are integers, satisfying the condition: $1 \leq x, y, z \leq n$. We  paint all the points of $T$ in such a way that: if the point $A(x_0, y_0, z_0)$ is painted then points $B(x_1, y_1, z_1)$ for which $x_1 \leq x_0, y_1 \leq y_0$ and $z_1 \leq z_0$ could not be painted. Find the maximal number of points that we can paint in such a way the above mentioned condition is satisfied.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left\lfloor \frac{3n^2 + 1}{4} \right\rfloor}
Let \( K \) be a field and consider \( a \in K \) with \( f(x) = x^n - a \) being an irreducible polynomial. If \( m \in \mathbb{N} \) is a divisor of \( n \), and \( \alpha \) is a root of \( f(x) \) in an extension field of \( K \), determine the minimal polynomial of \( \alpha^m \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^{n/m} - a}
Find all squares which have only two non-zero digits, one of them 3.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Determine all values of \( p \in \mathbb{R} \) for which \( |x-y|^p \) defines a metric on \( \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 1]}
Consider the set of complex numbers $z$ satisfying $|1+z+z^{2}|=4$. The maximum value of the imaginary part of $z$ can be written in the form $\frac{\sqrt{m}}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the value of $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
Miki has a dozen oranges of the same size and a dozen pears of the same size. Miki uses her juicer to extract 8 ounces of pear juice from 3 pears and 8 ounces of orange juice from 2 oranges. She makes a pear-orange juice blend from an equal number of pears and oranges. What percentage of the blend is pear juice? Express your answer as a percentage (%).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40%}
Find the fixed field of the automorphism $t \mapsto t-2$ of the field $k(t)$, where the characteristic of $k$ is non-zero.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k(t^p - t)}
Given the function $f(x)$, its domain is $\mathbb{R}$. When $x < 0$, $f(x) = x^5 - 1$; when $-1 \leq x \leq 1$, $f(-x) = -f(x)$; when $x > 0$, $f(x+1) = f(x)$. Calculate $f(2016)$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the limit without using L'Hôpital's rule:
\[ \lim_{a\to 0}\frac{\delta_{a}+\delta_{-a}-2\delta}{a^{2}} \]
where \(\delta_{a}\) is the Dirac distribution at the point \(a\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\delta''}
Evaluate the integral \( I = \int_{0}^{\pi/2} \lfloor \cot(x) \rfloor \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Determine the largest integer \( K \) such that player A can always place at least \( K \) red stones on a 20x20 grid, ensuring that the distance between any two red stones is never equal to \( \sqrt{5} \), regardless of how player B places blue stones on the grid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Given two independent random variables $X$ and $Y$, where $X \sim U[a, b]$ and $Y \sim U[c, d]$, with $a < c < 0$ and $d < b > 0$. How do you calculate the probability $P(X > Y)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{2b - c - d}{2(b-a)}\)}
Given the arithmetic sequence $\{a_n\}$, find the maximum number of different arithmetic sequences that can be formed by choosing any 3 distinct numbers from the first 20 terms.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{180}
Oleg drew an empty 50×50 table and wrote a number above each column and to the left of each row. It turned out that all 100 written numbers are different, 50 of which are rational and the remaining 50 are irrational. Then, in each cell of the table, he recorded the product of the numbers written near its row and its column (a "multiplication table"). What is the maximum number of products in this table that could turn out to be rational numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1250}
Suppose two independent sequences of Bernoulli trials continue until a success occurs. For sequences A and B, with respective success-probabilities a and b, what is the probability that A is shorter than B?

Here's an example: Suppose that I have two unfair coins. Coin 1 has a probability of coming up heads of 1/3, and coin 2 has a probability of coming up heads of 2/3. I flip both coins at the same time. What is the probability that Coin 1 is heads before coin 2?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/7}
Find the 2021st derivative of the function \( f(x) = \frac{1+x}{1+x+x^2} \) evaluated at \( x = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2021!}
12. Let $A, B$ be subsets of the set $X=\{1,2, \cdots, n\}$. If every number in $A$ is strictly greater than all the numbers in $B$, then the ordered subset pair $(A, B)$ is called "good". Find the number of "good" subset pairs of $X$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n}+n\cdot2^{n-1}}
Given three points \((a, b), (c, d), (x, y)\) on a circle with radius \(r\) in a rectangular coordinate plane, determine the minimum possible value of the expression:
\[
(ax + by - c)^2 + (bx - ay + d)^2 + (cx + dy + a)^2 + (dx - cy - b)^2.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2r^4 + 2r^2}
A trailer carries 3, 4, and 5 crates on a trip. Each crate weighs a certain amount. The maximum weight of the crates on a single trip is 6250 kg. What is the minimum weight of each crate?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1250}
Let  $ABC$  be a triangle with circumcircle  $\Gamma$  and let  $D$  be the midpoint of minor arc  $BC$ . Let  $E, F$  be on  $\Gamma$  such that  $DE \bot AC$  and  $DF \bot AB$ . Lines  $BE$  and  $DF$  meet at  $G$ , and lines  $CF$  and  $DE$  meet at  $H$ . Given that  $AB = 8, AC = 10$ , and  $\angle BAC = 60^\circ$ , find the area of  $BCHG$ .

*Note: this is a modified version of Premier #2*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{3}}
Determine the smallest positive integer $n$ such that in any two-coloring of a complete graph $K_n$, there exist $m$ pairwise disjoint edges of the same color.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3m - 1}
A graph has 12 vertices, and one of these vertices has a degree of 9. How many different labeled spanning trees can be formed that include this vertex?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5445}
There was an integer number of cheese heads in the kitchen. During the night, some rats came and ate 10 cheese heads, with each rat eating an equal portion. Afterward, some of the rats got sick from overeating. The next night, the remaining seven rats finished the remaining cheese, but each rat was able to eat only half as much cheese as they did the previous night. How much cheese was originally in the kitchen?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Shapovalov A.V.

At a round table, 12 places were prepared for the jury with each place labeled with a name. Nikolai Nikolaevich, who arrived first, absent-mindedly sat not in his own place, but in the next one clockwise. Each subsequent jury member, approaching the table, would take their own place or, if it was already occupied, walk around the table clockwise and sit in the first available place. The resulting seating arrangement of the jury members depends on the order in which they approached the table. How many different seating arrangements can arise?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1024}
# 7. Problem 7

An angle of 90 degrees is covered by some angles with the same vertex in two or three layers, and the sum of the angles is 290 degrees. Find the measure of the angle covered in three layers.

#
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Given Laila had a 10-hour work day and attended three different meetings during this period, the first meeting took 50 minutes, the second meeting twice as long as the first, and the third meeting half as long as the second meeting, calculate the percentage of her work day that was spent attending meetings.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33.33\%}
Find the largest natural number such that all its digits are distinct, and the sum of any two adjacent digits is a square of a natural number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6310972}
Determine the height of a tower from a 20-meter distant building, given that the angle of elevation to the top of the tower is 30° and the angle of depression to the base of the tower is 45°.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["20 \\left(1 + \\frac {\\sqrt {3}}{3}\\right)"]}
Calculate the remainder when $12^{34^{56^{78}}}$ is divided by 90.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Define $\lfloor x \rfloor$ as the greatest integer less than or equal to x, and ${x} = x - \lfloor x \rfloor$ as the fractional part of $x$. If $\lfloor x^2 \rfloor =2 \lfloor x \rfloor$ and $\{x^2\} =\frac12 \{x\}$, determine all possible values of $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$x \in \{0, 0.5, 1.5, 2\}$}
Find an upper bound for the recursive function \( T(n) = T(n-1) + n^2 + n^2 \log n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(n^3 \log n)}
A farmer has thrice as many pigs as cows, 7 more cows than goats, twice as many chickens as the total of cows and pigs, and half as many ducks as the sum of goats and chickens. If there are also 5 more sheep than the total number of cows and chickens combined, and the farmer has 346 animals total, how many goats does he have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given the function $f(x) = \frac{1}{2}x^{2} + a\ln x$, if for any two distinct positive real numbers $x_{1}$, $x_{2}$, it always holds that $\frac{f(x_{1})-f(x_{2})}{x_{1}-x_{2}} > 2$, then the range of the real number $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[1,+\infty)}
Determine the number of ways to arrange 10 men and 4 women around a circular table such that no three women sit consecutively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5356108800}
Consider coins with positive real denominations not exceeding 1 . Find the smallest $C>0$ such that the following holds: if we are given any 100 such coins with total value 50 , then we can always split them into two stacks of 50 coins each such that the absolute difference between the total values of the two stacks is at most $C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{50}{51}}
How many positive integers at most 420 leave different remainders when divided by each of 5, 6, and 7?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{250}
Calculate the fundamental group of the space defined by the equation: \[ X=\{(x,y,z)\in\mathbb{R}^3:(x^2+y^2+z^2-100)[(x-10)^2+y^2+z^2-1]=0\}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $ABC$ be an equilateral triangle with side length $1$. This triangle is rotated by some angle about its center to form triangle $DEF$. The intersection of $ABC$ and $DEF$ is an equilateral hexagon with an area that is $\frac{4}{5}$ the area of $ABC$. The side length of this hexagon can be expressed in the form $\frac{m}{n}$ 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{7}
Calculate the flux of the vector field \( \vec{F}(x,y,z) = \langle x, y, 3 \rangle \) out of the region \( T \) bounded by \( z = x^2 + y^2 \) and \( z = 4 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16\pi}
Tyler has two calculators, both of which initially display zero.  The first calculators has only two buttons, $[+1]$ and $[\times 2]$.  The second has only the buttons $[+1]$ and $[\times 4]$.  Both calculators update their displays immediately after each keystroke. 
A positive integer $n$ is called ambivalent if the minimum number of keystrokes needed to display $n$ on the first calculator equals the minimum number of keystrokes needed to display $n$ on the second calculator. Find the sum of all ambivalent integers between $256$ and $1024$ inclusive.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
34776
}
18th Putnam 1958 Problem A2 A rough sphere radius R rests on top of a fixed rough sphere radius R. It is displaced slightly and starts to roll off. At what point does it lose contact?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\cos^{-1}\frac{10}{17}}
Determine the total variation of the two-variable step function given by:
\[ f(x,y) = \begin{cases} 1 & \text{when } 0<x<a,\ 0<y<b; \\\ 0 & \text{otherwise} \end{cases} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2(a + b)}
In \(\Delta ABC\), the points \(P(1,1,2)\) and \(Q(4,7,8)\) satisfy \(\frac{AP}{AQ}=\frac{BP}{BQ}= \frac{CP}{CQ}=2\). Find the coordinates of the circumcenter of \(\Delta ABC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(5, 9, 10)}
What is the value of the infinite sum $$\sum_{n=1}^{\infty} \dfrac{1}{4n^2+16n+7}$$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{71}{630}\)}
Evaluate the limit of the sequence as \( n \to \infty \):

\[ a_n = n\left(\sqrt[n]{ea} - \sqrt[n]{a}\right), \quad n \ge 1, \ a > 0 \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the largest integer \( n \) such that for all integers \( b \) that are relatively prime to \( n \), it holds that \( b^{24} \equiv 1 \pmod{n} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{131040}
Determine the dimension of the subset of pairs of 2x2 matrices \((c_1, c_2)\) that generate a 2-dimensional Lie algebra, where the commutator \([c_1, c_2] = k_1 c_1 + k_2 c_2\) for some scalars \(k_1\) and \(k_2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Six people play the following game: They have a cube, initially white. One by one, the players mark an $ X$ on a white face of the cube, and roll it like a die. The winner is the first person to roll an $ X$ (for example, player 1 wins with probability $ \frac {1}{6}$, while if none of players 1-5 win, player 6 will place an $ X$ on the last square and win for sure). What is the probability that the sixth player wins?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\[ \frac{5}{324} \]}
Sue is traveling from New Orleans to San Francisco with multiple layovers and time zone changes. Her journey starts at New Orleans, where she takes a flight to Atlanta, then flies to Chicago, from there to New York, to Denver, and finally, from Denver to San Francisco. All layover times and flight durations are given as local times.

The New Orleans to Atlanta flight takes 2 hours. Atlanta is in the same time zone as New Orleans. Sue has a 4-hour layover in Atlanta before the flight to Chicago. The Atlanta to Chicago flight takes 5 hours. Chicago is 1 hour behind Atlanta. Sue has a 3-hour layover in Chicago. Her last flight before New York, Chicago to New York, takes 3 hours. New York is 1 hour ahead of Chicago. When she arrives in New York, Sue must wait 16 hours for her next flight.

From New York, Sue first travels to Denver, which takes 6 hours. Denver is 2 hours behind New York. Sue has a 5-hour layover in Denver before her final leg to San Francisco. The flight from Denver to San Francisco takes 4 hours. San Francisco is 1 hour behind Denver.

Calculate the total time Sue took to reach San Francisco from New Orleans, considering the time zones and time differences affecting her travel and layovers along her entire journey.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Calculate \( \mathbb{E}\left[\left(\int_{0}^{\infty}g(t)\,dW_t\right)^2\right] \) where \( g(t) = (W_3 - W_2)\mathbf{1}_{[3,4)}(t) + (W_4 - W_2)\mathbf{1}_{[4,6)}(t) \) and \((W_t)_{t \ge 0}\) is a Brownian motion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find the angle at which the parabola \(y^{2}=2px\) intersects the circle \(\left(x-\frac{p}{2}\right)^{2}+y^{2}=4p^{2}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{3}}
11. Given that when $|x| \leqslant 1$, we always have
$$
2 p x^{2}+q x-p+1 \geqslant 0 \text {. }
$$

Find the maximum value of $p+q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["2"]}
Consider a sequence of positive integers $a_1, a_2, a_3, \ldots$ such that $\gcd(a_m, a_n) > 1$ if and only if $|m - n| = 1$. Determine the sum of the four smallest possible values of $a_2$. \(\boxed{\text{Your answer here}}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Let  $f(x)=\cos(\cos(\cos(\cos(\cos(\cos(\cos(\cos(x))))))))$ , and suppose that the number  $a$  satisfies the equation  $a=\cos a$ . Express  $f'(a)$  as a polynomial in  $a$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - 4a^2 + 6a^4 - 4a^6 + a^8}
For a given positive integer \( n \geq 2 \), determine the largest positive real number \( \lambda(n) \) such that for any real numbers \( x_1, x_2, \ldots, x_n \), not all of which are positive, the inequality holds:
\[
\prod_{i=1}^{n}\left(x_{i}^{2}-x_{i}+1\right) \geq \lambda(n) \cdot \left(\left(\prod_{i=1}^{n} x_{i}\right)^{2}-\prod_{i=1}^{n} x_{i}+1\right)
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left( \dfrac{3}{4} \right)^{n-1}}
Determine the dimension of the vector space \( \mathbb{C}[x,y]/I \) over \( \mathbb{C} \), where \( I = \langle x^2 + 4x + 4, xy+x+2y+2, y^3 + 3y^2 + 3y + 1 \rangle \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Compute

$3(1+3(1+3(1+3(1+3(1+3(1+3(1+3(1+3(1+3)))))))))$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{88572}
What is the last two digits of the decimal representation of  \( {9}^{{8}^{{7}^{{.}^{{.}^{{.}^{2}}}}}}  \) ? 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21
}
A rope is folded into 3 equal parts, then folded in half again, and cut in the middle. In total, it can be cut into \_\_\_\_\_\_ segments.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
For each positive integer $ n$, let $ f(n)$ denote the greatest common divisor of $ n!+1$ and $ (n+1)!$. Find, without proof, a formula for $ f(n)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1

}