inclusionAI/Ring-lite-rl-data huggingface dataset:
In triangle $ABC$, segment $AD$ is the angle bisector, $AD = l$, $AB = c$, and $AC = b$. Find angle $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \arccos \frac{(b+c) l}{2 b c}}
Given four points \( A, B, C, D \) on the surface of a unit sphere, such that \( AB = AC = AD \) and \( BC = CD = DB \), find the maximum volume of the tetrahedron \( ABCD \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{8 \sqrt{3}}{27}}
of the 80 people in a room , 2 / 5 are women . if 1 / 2 of the people are married , what is the maximum number of women in the room who could be unmarried ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
In an exam, 153 people scored no more than 30 points, with an average score of 24 points. 59 people scored no less than 80 points, with an average score of 92 points. The average score of those who scored more than 30 points is 62 points. The average score of those who scored less than 80 points is 54 points. How many people participated in this exam?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1007}
Consider two sequences \(a_n\) and \(b_n\) where \(a_n, b_n > 0\). If \(a_n \sim b_n\) and both \(\sum a_n\) and \(\sum b_n\) diverge, determine whether it implies that \(\lim_{N\rightarrow\infty} \frac{\sum_{n\leq N} a_n}{\sum_{n\leq N} b_n} = 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In Ms. Smith's class, each student averages one day absent out of thirty. What is the probability that out of any two students chosen at random, one student will be absent while the other is present? Express your answer as a percent rounded to the nearest tenth.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6.4}
Determine if $x = 2$ is the only real solution to the equation $a^x + b^x = c^x$ when $(a, b, c)$ is a Pythagorean triplet.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given a finitely generated module $M$ over a commutative ring $A$, suppose there exist a positive integer $n$ and homomorphisms $\pi: A^n \rightarrow M$ (surjective) and $\phi: A^n \rightarrow M$ (injective). Does it follow that $M \cong A^n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{M \cong A^n}
Determine whether the limit of the series \( \sum_{k \ge 0} \frac{\Gamma(j)}{\Gamma(j+k/2)}(-t)^k \) approaches zero as \( t \to \infty \), where \( j \in \mathbb{N} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A hollow glass sphere with uniform wall thickness and an outer diameter of $16 \mathrm{~cm}$ floats in water in such a way that $\frac{3}{8}$ of its surface remains dry. What is the wall thickness, given that the specific gravity of the glass is $s = 2.523$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["0.8"]}
The length of a rectangle is 5 less than 3 times its width. Find the dimensions of the rectangle if the area is 782 m².
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{46 \text{ m} \times 17 \text{ m}}
Consider all possible quadrilaterals $ABCD$ that have the following properties; $ABCD$ has integer side lengths with $AB\parallel CD$, the distance between $\overline{AB}$ and $\overline{CD}$ is $20$, and $AB = 18$. What is the maximum area among all these quadrilaterals, minus the minimum area?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2130}
Let $ABCD$ be a quadrilateral with $\overline{AB}\parallel\overline{CD}$, $AB=16$, $CD=12$, and $BC<AD$. A circle with diameter $12$ is inside of $ABCD$ and tangent to all four sides. Find $BC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Determine the number of zeros of the polynomial \( p(z) = 1 + 2z^4 + \frac{7}{10}z^{10} \) that lie within the unit disc, and specify their multiplicities.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Quadrilateral ABCD is a cyclic quadrilateral inscribed in circle O with diameter BC. Point D is the midpoint of arc AC. Given that AB = 2 and AD = 2√3, find the area of circle O. Express your answer in terms of π as a product of a numerical value and π.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9 π}
Given a quadratic polynomial \(P(x,y)=ax^2+bxy+cy^2+ex+fy+g\) with integer coefficients, where \((a,b,c)=1\), determine if there exists an equivalent polynomial \(P'(x,y)=a'x^2+b'xy+c'y^2+e'x+f'y+g'\) such that \((b',a'c')=1\). Answer 'yes' or 'no'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{yes}
In a certain company, there are 100 shareholders, and any group of 66 of them collectively owns at least 50% of the company's shares. What is the maximum percentage of all shares that one shareholder can own? (The percentage of the company's shares owned by a shareholder can be a non-integer value.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25\%}
Kelvin the Frog writes 2020 words on a blackboard, with each word chosen uniformly randomly from the set $\{\verb|happy|, \verb|boom|, \verb|swamp|\}$. A multiset of seven words is merry if its elements can spell $``\verb|happy happy boom boom swamp swamp swamp|."$ For example, the eight words
\[\verb|swamp|, \verb|happy|, \verb|boom|, \verb|swamp|, \verb|swamp|, \verb|boom|, \verb|swamp|, \verb|happy|\]contain four merry multisets. Determine the expected number of merry multisets contained in the words on the blackboard.

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\[ \frac{70 \cdot \dbinom{2020}{7}}{729} \]}
Jeremy wrote all the three-digit integers from 100 to 999 on a blackboard. Then Allison erased each of the 2700 digits Jeremy wrote and replaced each digit with the square of that digit. Thus, Allison replaced every 1 with a 1, every 2 with a 4, every 3 with a 9, every 4 with a 16, and so forth. The proportion of all the digits Allison wrote that were ones is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{87}
There are many CDs in the store. The rock and roll CDs are $7 each, the pop CDs are $12 each, the dance CDs are $5 each, and the country CDs are $9 each. If Julia buys 3 or more of each genre, she gets a 15% discount on that genre. She wants to buy 5 rock and roll CDs, 3 pop CDs, 6 dance CDs, and 4 country CDs, but some genres have limited availability: there are only 4 rock and roll CDs left, and only 5 dance CDs left. Julia only has 80 dollars. How much money is she short?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$26.25}
Three people, A, B, and C, visit three tourist spots, with each person visiting only one spot. Let event $A$ be "the three people visit different spots," and event $B$ be "person A visits a spot alone." Then, the probability $P(A|B)=$ ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Calculate the number of different spanning trees for the simple graph $G_{k,n}=([n],E)$, where $ij\in E\Leftrightarrow 0 <|i-j|\leq k$ for $i\neq j\in [n]$. Specifically, find the number of spanning trees when $k=3$ and $n=5$. Use Kirchoff's Matrix Tree theorem to verify your result.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}
At an international table tennis championship, 200 participants arrived. The tournament follows a single-elimination format, meaning that in each match two players compete, the loser is eliminated from the championship, and the winner remains. Find the maximum possible number of participants who have won at least three matches.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{66}
Find all positive integers \( a,b,c \) and prime \( p \) satisfying that
 \( {2}^{a}{p}^{b} = {\left( p + 2\right) }^{c} + 1. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
a=1, b=2, c=2, p=5
}
Can the Gaussian integral \( \int_{-\infty}^{\infty} e^{-x^2}\,dx \) be evaluated using contour integration techniques from complex analysis?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\pi}}
A man named Juan has three rectangular solids, each having volume 128. Two of the faces of one solid have areas 4 and 32. Two faces of another solid have areas 64 and 16. Finally, two faces of the last solid have areas 8 and 32. What is the minimum possible exposed surface area of the tallest tower Juan can construct by stacking his solids one on top of the other, face to face? (Assume that the base of the tower is not exposed).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{688}
Let $P_n$ be the number of permutations $\pi$ of $\{1,2,\dots,n\}$ such that\[|i-j|=1\text{ implies }|\pi(i)-\pi(j)|\le 2\]for all $i,j$ in $\{1,2,\dots,n\}.$  for $n\ge 2,$ the quantity\[P_{n+5}-P_{n+4}-P_{n+3}+P_n\]does not depend on $n,$  find its value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{The value is 4.}
In triangle $ABC$, the angle bisector $AM$ and the median $BN$ intersect at point $O$. It turns out that the areas of triangles $ABM$ and $MNC$ are equal. Find $\angle MON$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
Given $n=2020$, sort the 6 values: $$n^{n^2},\, 2^{2^{2^n}},\, n^{2^n},\, 2^{2^{n^2}},\, 2^{n^n},\, \text{and}\, 2^{n^{2^2}}$$ from least to greatest. Provide your answer as a 6-digit permutation of the string "123456", where the number $i$ corresponds to the $i$-th expression in the list, from left to right.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{163542}
A retailer sold a watch at a loss of 10% in USD. On the same day, they sold an identical watch for a gain of 4% in EUR. If they had sold the first watch for 200 USD more, there would have been a gain of 4% in USD as well. On that particular day, the exchange rate was changing: the USD was depreciating at a rate of 5% in a 24-hour period, while the EUR was appreciating at a rate of 3% during the same time frame. Assume the watches were sold at exactly noon, halfway through the 24-hour period, and that the exchange rate changes are linear. Calculate the original combined cost price of both watches in USD and EUR at the time of purchase according to these parameters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1428.57}
The current birth rate per certain number of people is unknown, whereas the corresponding death rate is 16 per the same number of people. If the net growth rate in terms of population increase is 1.2 percent, and there are initially 3000 persons, what is the birth rate per certain number of people?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52}
In the convex polygon $M N K L P$, the segment $N L$ bisects each of the angles $K N P$ and $K L M$, and the segment $K P$ bisects each of the angles $M K L$ and $N P L$. The diagonal $N P$ intersects the diagonals $M K$ and $M L$ at points $F$ and $E$, respectively. Determine if $K F = L E$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{KF = LE}
Assuming that each distinct arrangement of the letters in $DISCUSSIONS is equally likely to occur, what is the probability that a random arrangement of the letters in $DISCUSSIONS has all the $S$’s together?



Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$ \frac{4}{165} $}
Teddy owns a square patch of desert. He builds a dam in a straight line across the square, splitting the square into two trapezoids. The perimeters of the trapezoids are$ 64$ miles and $76$ miles, and their areas differ by $135$ square miles. Find, in miles, the length of the segment that divides them.

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Given that $4^{-1} \equiv 57 \pmod{119}$, find $64^{-1} \pmod{119}$, as a residue modulo 119. (Give an answer between 0 and 118, inclusive.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
18. A circle with center $C$ and radius $r$ intersects the square $E F G H$ at $H$ and at $M$, the midpoint of $E F$. If $C, E$ and $F$ are collinear and $E$ lies between $C$ and $F$, what is the area of the region outside the circle and inside the square in terms of $r$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{r^{2}(\frac{22}{25}-\frac{\tan^{-1}(4/3)}{2})}
Determine whether there is a closed-form solution for the nonlinear ordinary differential equation \( y'' = y^3 - y \) with the initial conditions \( y(0) = 1 \) and \( y'(0) = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y(x) = 1}
Determine whether the sequence \( y_{n} = n\ln(n) - n - (\ln(n))^{2} \) converges or diverges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Determine whether the sequence \( a_{n} = \frac{1}{\sqrt{n^2+n}} + \frac{1}{\sqrt{n^2+n+1}} + \ldots + \frac{1}{\sqrt{n^2+2n-1}} \) converges to a limit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A Gauss brand light bulb will work for 24999 hours. If it is used for exactly 2 hours every day starting on a Monday, on what day of the week will it stop working? Express your answer as the name of the day of the week.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Friday}
Consider the function \( f(x) = \frac{1}{3}(x^2 + x + 1) \). Define \( f_0(x) = x \) and for all \( k \ge 1 \), \( f_k(x) = f(f_{k-1}(x)) \). Let \( a_k = f_k(0) - f_{k-1}(0) \). Determine whether the series \( \sum\limits_{k=1}^{\infty} k a_k \) is infinite.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Let $G$ be an infinite subgroup of $(\mathbb{C} \setminus \{0\}, \cdot)$ such that $G$ is compact as a subset of $\mathbb{C}$. Determine whether it is true that $G = S^1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{G = S^1}
Let $a\in\mathbb R$, define a sequence $a_n$ as the distance from $a$ to the nearest number of the form $\frac{m}{2^n}$, where $m\in\mathbb Z$. Find the maximal value of the sum $\sum\limits_{n=0}^{\infty}a_n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{2}{3}$}
From among $2^{1/2},$ $3^{1/3},$ $8^{1/8},$ $9^{1/9}$ those which have the greatest and the next to the greatest values, in that order, are?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$3^{1/3}, 2^{1/2}$
}
The graphs of the equations 
$y=k, \qquad y=\sqrt{3}x+2k, \qquad y=-\sqrt{3}x+2k,$ 
are drawn in the coordinate plane for $k=-10,-9,-8,\ldots,9,10.\,$ These $63$ lines cut part of the plane into equilateral triangles of side length $\tfrac{2}{\sqrt{3}}.\,$ How many such triangles are formed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{660}
Buses leave Moscow for Voronezh every hour, at 00 minutes. Buses leave Voronezh for Moscow every hour, at 30 minutes. The trip between cities takes 8 hours. How many buses from Voronezh will a bus leaving Moscow meet on its way?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
How many positive integers less than $250$ are multiples of $5$, but not multiples of $10$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Determine if the expression \( \frac{d}{dy}\bigg|_{y = x}\big[[f'(y) - f'(x)]^2\big] \) is defined, given that \( f : S \subseteq \mathbb{R} \to \mathbb{R} \) is differentiable on \( S \) and \( f' \) is also differentiable on \( S \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Ashraful has 12 coins, each of which is a 5-cent coin or a 10-cent coin. There are exactly 17 different values that can be obtained as combinations of one or more of her coins. How many 10-cent coins does Ashraful have?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5
}
A square with side length 8 is cut in half, creating two congruent rectangles. What are the dimensions of one of these rectangles?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4 \times 8}
What is the value of

$$\log_37\cdot\log_59\cdot\log_711\cdot\log_913\cdots\log_{21}25\cdot\log_{23}27?$$ 


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
If E is a set with infinite measure and c is a positive constant function defined on E, what is the value of the integral of c over E in terms of c and the measure M(E)? Specifically, how should this be interpreted when M(E) is infinite?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
$\triangle ABC$ has positive integer side lengths of $x$,$y$, and $17$. The angle bisector of $\angle BAC$ hits $BC$ at $D$. If $\angle C=90^\circ$, and the maximum value of $\frac{[ABD]}{[ACD]}=\frac{m}{n}$ where $m$ and $n$ are relatively prime positive intgers, find $m+n$. (Note $[ABC]$ denotes the area of $\triangle ABC$).


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{162}
Let \( H \) be a Hilbert space with an orthonormal basis \((e_n)_{n \in \mathbb{N}}\) and \( F \) a linear operator such that \( \langle e_k, F e_n \rangle = \phi(n,k) \). Find an estimate for the operator norm \( \lVert F \rVert \) in terms of \( \phi(n,k) \). Apply this estimate to the case where \( \phi(n,k) = \frac{1}{n+k} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Determine whether the ideal \((I, J)\) is equal to the ideal \((I, J, IJ)\), where \(IJ\) is the product of the ideals \(I\) and \(J\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(I, J) = (I, J, IJ)}
Consider a probability measure without atoms. Does the probability of a sequence of nested sets approaching an empty set tend to 0?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
6. The first circle with center at point $O$ is inscribed in triangle $A B C$. Points $A$ and $B$ lie on the second circle with the same center $O$. Line $A C$ intersects the second circle at point $D$ $(D \neq A)$, and line $B C$ intersects the second circle at point $E(B \neq E)$. It is known that angle $A B C$ is equal to angle $C A E$. Find the cosine of angle $B A C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1+\sqrt{5}}{4}}
Evaluate whether the expression \( \epsilon^{\rho\sigma\mu\nu}\epsilon_{\rho\sigma\mu\nu} \) is zero, given that \( \epsilon^{\rho\sigma\mu\nu}\epsilon_{\mu\nu\rho'\sigma'} = -2(\delta^{\rho}_{\rho'}\delta^{\sigma}_{\sigma'}-\delta^\rho_{\sigma'}\delta^{\sigma}_{\rho'}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-24}
In a cross-country race, each team has 5 players. If a player earns $n$ points for their team by placing $n$-th, the team with fewer points wins. Without any ties, what is the number of possible scores for the winning team?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
At the 2013 Winnebago County Fair a vendor is offering a "fair special" on sandals. If you buy one pair of sandals at the regular price of \$50, you get a second pair at a 40% discount, and a third pair at half the regular price. Javier took advantage of the "fair special" to buy three pairs of sandals. What percentage of the \$150 regular price did he save?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Find all values of $k$ for which the system
\begin{align*}
x + ky - z &= 0, \\
kx - y - z &= 0, \\
x + y - kz &= 0
\end{align*}has a non-trivial solution.  (In other words, find all values of $k$ for which the system has a solution other than $(x,y,z) = (0,0,0).$)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1,0,1}
Consider the series \(\sum_{i=1}^\infty c_i^{(0)}\) which is a positive convergent series. Define the series \(\sum_{i=1}^\infty c_i^{(1)} = \sum_{i=1}^\infty \ln(1+c_i^{(0)})\). Generally, let \(c_i^{(N+1)} = \ln(1+c_i^{(N)})\) and define the sum \(S^{(N)} = \sum_{i=1}^\infty c_i^{(N)}\). Determine the nature of the limit \(\lim_{N\rightarrow\infty} S^{(N)}\). Is this limit zero or nonzero?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Can the squares of a $1990 \times 1990$ chessboard be colored black or white so that half the squares in each row and column are black and cells symmetric with respect to the center are of opposite color?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$NO$}
Determine the asymptotic behavior of the series \( \sum_{k=1}^n \frac{1}{\sqrt{k}} \) and compare it to the series \( \sum_{k=1}^n \frac{1}{k} \), which is approximately \( \log(n) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{n}}
Suppose you have a $k \times n$ matrix $A$ where $k \geq n$ and $\mathrm{rank}(A) = m$ with $m \leq n \leq k$. Let $r_1, \dots, r_k$ be the rows of $A$. Consider the $n \times n$ matrix $B = r_1^Tr_1 + \cdots + r_k^Tr_k$. Determine if $\mathrm{rank}(B) = m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m}
Evaluate the line integral over the curve defined by the intersection of the cone \(z^2 = x^2 + y^2\) and the cylinder \(x^2 + y^2 = 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi}
For any integer $r \geq 1$, determine the smallest integer $h(r) \geq 1$ such that for any partition of the set $\{1, 2, \cdots, h(r)\}$ into $r$ classes, there are integers $a \geq  0 \ ; 1 \leq x \leq y$, such that $a + x, a + y, a + x + y$ belong to the same class.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$h(r) = 2r$

}
A friendly football match lasts 90 minutes. In this problem, we consider one of the teams, coached by Sir Alex, which plays with 11 players at all times.  
 Sir Alex wants for each of his players to play the same integer number of minutes, but each player has to play less than 60 minutes in total. What is the minimum number of players required?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Five fair six-sided dice are rolled. What is the probability that at least three of the five dice show the same value?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{23}{108}}
Two trains of different lengths are running towards each other on parallel lines at 42 kmph and 30 kmph respectively. The trains will be clear of each other in 12.998960083193344 seconds from the moment they meet. If the second train is 160 m long, how long is the first train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{99.98}
Folklore
In Chicago, there are 36 gangsters, some of whom are at odds with each other. Each gangster is a member of several gangs, and there are no two gangs with the same membership. It turned out that gangsters who are in the same gang do not feud, but if a gangster is not a member of a certain gang, then he feuds with at least one of its members. What is the maximum number of gangs that could exist in Chicago?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3^{12}}