meetforever/GRPO_CL huggingface dataset:
user: You are the proud owner of the new Multispecies Animal Protection Sanctuary (MAPS). As you arrive on the morning of the grand opening, you realize with horror that some of the animals are in the wrong enclosures. You will deal with the overnight zookeeper later, but right now you need to move all of the animals back to their proper areas before the zoo opens.

You may choose to begin at any of the enclosures, and you must move the animals, one at a time, back to their respective areas. Since time is of the essence, you cannot travel from one enclosure to another unless you are moving an animal directly back to its proper area. Note that the zoo is laid out in such a fashion that it is always possible to travel from one enclosure to another without passing through any other enclosures.

Determine if it is possible to return all of the animals back to their proper enclosures subject to these restrictions.

-----Input-----
The first line of input contains two space-separated integers, $n$ and $m$, where $1\leq n\leq 10000$ denotes the number of different enclosures and $1\leq m\leq 50000$ denotes the total number of animals in the zoo. Each of the next $n$ lines gives the starting state of a different enclosure. Each enclosure is described on a single line using a series of space-separated tokens. The first token is a string indicating which type of animal belongs in this particular enclosure. The second token is a non-negative integer $a$, identifying the number of animals currently in this enclosure. This is followed by $a$ tokens on the same line, each of which is a string indicating the type of one of the animals currently in this enclosure.

Each animal type is composed of lowercase letters (a–z) and contains between $1$ and $8$ characters, inclusive. Each animal in the zoo belongs to exactly one enclosure, and that may or may not correspond to the enclosure that the animal is currently in.

-----Output-----
If at least one animal is not in its proper enclosure, output “POSSIBLE” if it is possible to restore all of the animals back to their proper locations under the above restrictions; otherwise, output “IMPOSSIBLE”. If all the animals are already in their proper enclosures, output “FALSE ALARM” instead.

-----Examples-----
Sample Input:
3 6
monkey 2 lion penguin
lion 3 monkey penguin lion
penguin 1 monkey
Sample Output:
POSSIBLE
user: Generating a random number sequence is not easy. Many sequences may look random but upon closer inspection we can find hidden regularities in the occurrence of the numbers. For example, consider the following $100$-digit “random” sequence starting with $4, 7, 9, \ldots $:

  4 7 9 5 9 3 5 0 0 1 7 8 5 0 2 6 3 5 4 4 4 6 3 3 2 7 1 8 7 8 7 6 1 1 7 2 5 4 7 2 0 4 4 5 8 3 0 6 9 3 2 6 6 8 5 2 5 1 2 7 2 4 1 0 0 4 9 1 8 7 5 0 4 4 8 4 3 2 6 8 8 5 6 7 0 9 7 0 3 6 1 4 4 1 2 3 2 6 9 9

If you look closely, whenever a $4$ is followed immediately by another $4$, the third value after this will be a $3$ (we’re sure you saw that). We’ll call features like this triple correlations and we’ll represent it as $4(1)4(3)3$, where the numbers in parentheses indicate how far away each number on either side of the parentheses must be. More precisely, a sequence of $p$ digits has an $a(n)b(m)c$ triple correlation if
 - any time there is an $a$ followed $n$ locations later by a $b$ there is always a $c$ located $m$ locations after the $b$ unless $b$’s location is within distance $m-1$ of the end of the sequence.
 - any time there is a $b$ followed $m$ locations later by a $c$ there is always an $a$ located $n$ locations before the $b$ unless $b$’s location is within distance $n-1$ of the beginning of the sequence.
 - any time there is an $a$ followed $n+m$ locations later by a $c$ there is always a $b$ located $n$ locations after the $a$.
 - the correlation occurs at least $\lceil p/40 \rceil + 1$ times, where $\lceil x \rceil $ denotes the smallest integer greater than or equal to $x$.

Such correlations are tough to spot in general, so that’s where we need your help. You will be given a sequence of digits and must search it for triple correlations.

-----Input-----
Input starts with a positive integer $p$ ($p \leq 1000$) indicating the number of random digits in the sequence. Starting on the next line will be the $p$ digits, separated by spaces and potentially spread across multiple lines. No line will contain more than $100$ digits and there will be no blank lines.

-----Output-----
Display triple correlation a(n)b(m)c found if it occurs in the list (with the appropriate values for a, b, c, n, and m filled in) or the phrase random sequence otherwise. If there are multiple triple correlations in the sequence, display the one which begins earliest in the sequence. If there is still a tie, choose the one with the smaller value of n, and if there is still a tie, choose the one with the smaller value of m.

-----Examples-----
Sample Input:
100
4 7 9 5 9 3 5 0 0 1 7 8 5 0 2 6 3 5 4 4
4 6 3 3 2 7 1 8 7 8 7 6 1 1 7 2 5 4 7 2
0 4 4 5 8 3 0 6 9 3 2 6 6 8 5 2 5 1 2 7
2 4 1 0 0 4 9 1 8 7 5 0 4 4 8 4 3 2 6 8
8 5 6 7 0 9 7 0 3 6 1 4 4 1 2 3 2 6 9 9
Sample Output:
triple correlation 4(1)4(3)3 found
user: The $N$ citizens of Eagleton have chosen to build their houses in a row along a single street (so that the houses can be numbered from $1$ to $N$, with $1$ the leftmost house and $N$ the rightmost). The houses have varying heights, with house $i$ being $h_ i$ inches tall. 

Unfortunately, all is not well in Eagleton: the citizens have a bit of an envy problem. Every day, one random citizen (the owner of house $i$, let’s say) emerges from their house and compares their house’s height to the heights of the two neighboring houses. If house $i$ is at least as tall as the average, plus $k$ inches (in other words, if $h_ i \geq (h_{i-1} + h_{i+1}) / 2 + k$), the citizen retreats back into their house, satisfied. Otherwise, the citizen remodels their house to have new height $(h_{i-1} + h_{i+1}) / 2 + k$. (The citizen does this remodeling even if the new height is only a tiny fraction of an inch taller than the old height—like we said, Eagleton has an envy problem.)

The left of house $1$ and the right of house $N$ is a nature preserve; the citizens of these houses treat the preserve as a “house” having height zero inches, for the purposes of the above calculations.

The city council of Eagleton is fed up with the constant construction traffic and noise, and has hired you to compute what Eagleton will look like when all of the remodeling is finally over. After some calculations, you discover that it is guaranteed that each house will converge to a final finite height after infinitely many days of the above remodeling process. Print the final height of the house that ends up tallest.

-----Input-----
The first line of input consists of an integer $N$ and real number $k$, separated by a space $(1 \leq N \leq 100000; 0 \leq k \leq 10^{20})$: the number of houses in Eagleton and the number of inches each citizen wants their own house to be taller than the average of their neighbors.

$N$ lines follow. The $i$th such line (starting at $i=1$) contains a real number $h_ i$ ($0 \leq h_ i \leq 10^{20}$), the initial height (in inches) of house $i$.

The real numbers in the input may be provided in scientific notation (see for instance Sample Input 2) and will have at most ten digits after the decimal point.

-----Output-----
Print the height (in inches) of the tallest house, after all houses have been remodeled over infinitely many days of the process described above. Your answer will be judged correct if the absolute or relative error is within $10^{-6}$.

-----Examples-----
Sample Input:
3 1
39
10
40
Sample Output:
40.5
user: For several years now, the Nordic Conference on Partitions and Combinatorics, NCPC, has had a growing number of participants. This year the organizing team is expecting an all time high record in the hundreds. Due to the politics of arranging this prestigious event, the conference site was decided a long time ago to be the Grand Hôtel in Stockholm. The hotel has two large dining halls, but unfortunately, each of these halls alone can only fit up to two thirds of the NCPC participants, so the participants are going to have to be divided in two groups.

This constraint calls for some thinking on behalf of the organizing team for the conference dinner: could they come up with some division of the participants in two parts, none of which is larger than $2/3$ of the entire group, meeting some witty division rule suitable for the occasion, which they could tell the participants for their amusement? After all, as long as there is some grand logic rule to which of the two dining halls you are being seated in, you (as a mathematician) would be happy! They thought for a while and came up with the following idea for the division: Is there a year $Y$ and a division of the participants in two parts such that every pair in the first part met for the first time some time before year $Y$, and every pair in the second part met for the first time some time in or after year $Y$? Now this clearly qualified as an appropriate rule to all of them, but the question was whether it would be possible.

-----Input-----
The first line of input contains an integer $4 \leq n \leq 400$, the number of participants, and $c$, the number of known first encounters. The next $c$ lines are each in the format $a$ $b$ $y$, meaning participants $a$ and $b$ ($1 \leq a < b \leq n$) met for the first time in year $y$ ($1948 \leq y < 2008$). No pair of participants will appear more than once on the list, and every pair of participants not in the list is assumed to have met only now (in the year $2008$).

-----Output-----
Output either the smallest year $Y$ such that it is possible to divide the participants in two parts, neither of which contains more than $2n/3$ people, such that all people in the first part first met before year $Y$, and all people in the second part first met in or after year $Y$. If there is no such year, output the string ’Impossible’.

-----Examples-----
Sample Input:
4 6
1 2 1987
2 3 1987
1 3 1987
2 4 1987
1 4 1987
3 4 1987
Sample Output:
Impossible
user: As you are probably aware, flight pricing can sometimes be surprisingly complex. For example, it can often be cheaper to take a much longer flight with several legs instead of flying directly between two airports. One of the reasons pricing seems to be complex is that airlines often try to obfuscate exactly how pricing works, in order to ensure that their customers choose more expensive flights.

One particular airline has deciced to take this obfuscation to the next level; they do not even offer an automated search of their flights. Instead, they describe their flights in a very peculiar format. For every one of their $N$ airports (which are numbered between $0$ and $N - 1$), they list either:
 - what airports they travel to from this airport, or
 - what airports they do not travel to from this airport.

.

To compensate for this complexity, the airline sets the price of every direct flight between two airports to the same amount.

Can you write a program that, given the descriptions of all the flights the airline provides, determine the minimum number of flights required to travel from airport $s$ to airport $t$?

-----Input-----
The first line of input contains an integer $1 \le N \le 10^5$, the number of airports, and the two integers $s$ and $t$ ($0 \le s, t < N$, $s \neq t$).

The next $N$ lines each describe the outgoing flights of an airport, starting with airport $0$. The line starts with a letter. If this letter is N, you will get a list of all destination airports from this airport. If this letter is C, you will get a list of all airports that are not destinations from this airport.

Following this letter is an integer $m$, the number of airports in the list. Finally, there will $m$ unique numbers $a_ i$ ($0 \le a_ i < N$) on the line, the airports in the list.

The sum of $m$ over all airports is at most $2 \cdot 10^5$.

-----Output-----
Output a single integer, the minimum number of flights required to travel from airport $s$ to airport $t$.

If there is no path, output “impossible”.

-----Explanation of Sample Input 1-----
The only flight from airport $0$ is to airport $2$. From airport $2$, there is also only a single flight going to airport $3$. From airport $3$, you can fly to any airport except airport $1$.

Since no airport has a direct flight to airport $1$, there cannot be any possible flight plan from airport $0$ to $1$, so the answer is impossible

-----Explanation of Sample Input 2-----
The only flight from airport $0$ is to airport $2$. From airport $2$, there is also only a single flight going to airport $3$. From airport $3$, you can fly to any airport except airport $0$.

Thus, there is a flight plan from $0$ to $1$ going from $0$ to $2$, to $3$, to $1$, which results in 3 flights. This is also the shortest flight plan.

-----Examples-----
Sample Input:
4 0 1
N 1 2
C 1 2
N 1 3
C 1 1
Sample Output:
impossible
user: You are teaching a class in programming, and you want to cover balanced parentheses. You’ve got a great visual aid, a sign with a very long, balanced string of parentheses. But, alas, somehow, your visual aid has been broken into pieces, and some pieces may be missing! You’ve got to try to put it back together as best you can. Given the string of parentheses on each piece, what is the longest balanced string you can form by concatenating some of them in some order? Each piece may be used at most once, and the pieces cannot be reversed.

A balanced string of parentheses is defined as:
 - The empty string
 - $AB$ where $A$ and $B$ are both balanced strings of parentheses
 - ($A$) where $A$ is a balanced string of parentheses

-----Input-----
Each input will consist of a single test case. Note that your program may be run multiple times on different inputs. The first line of input will contain a single integer $n$ ($1 \le n \le 300$), which is the number of pieces.

Each of the next $n$ lines will hold a single string $s$ ($1 \le |s| \le 300$), which consists only of the characters ’(’ and ’)’. This describes one of the pieces.

-----Output-----
Output a single integer, which is the length of the longest string of balanced parentheses you can form from the pieces. Note that the empty string is technically a balanced string of parentheses, so it is always possible to form a string of length at least $0$ (although the empty string is not a very effective visual aid!).

-----Examples-----
Sample Input:
3
())
((()
)()
Sample Output:
10
user: A unicyclic graph is a graph with exactly one cycle. A spanning subgraph of a graph $G$ is a subgraph that has one component and includes all the vertices of $G$. Given a simple graph $G$, count the number of spanning unicyclic subgraphs. The illustration below shows the visualization of Sample Input/Output $1$.

-----Input-----
The first line of the input contains two integers, $V$ and $E$, representing the number of vertices and edges of the graph $G$ respectively. ($1 \leq V \leq 17, 0 \leq E \leq V(V-1)/2$.)

The following $E$ lines each contains two integers $A_ i$ and $B_ i$, representing an edge $(A_ i, B_ i)$. It is guaranteed that $1 \leq A_ i < B_ i \leq V$ and as the graph is simple, no two pairs represent the same edge.

-----Output-----
Output one integer, representing the number of spanning unicylic subgraphs. As the number can be rather big, output it modulo $10^9 + 7$.

-----Examples-----
Sample Input:
4 5
1 2
1 3
2 3
1 4
2 4
Sample Output:
5
user: Fiona is an expert climber. She often brings some pegs with her, which she nails in some strategic places in the rock wall, so that less experienced climbers can use them for support. Fiona can climb to anywhere in the wall, but hammering a peg needs some balance, so she can only place a peg if she is standing in currently placed pegs (or, of course, the floor). She can remove a peg at any time and reuse it later. For each wall she is planning to visit, she has a careful plan for how to place and remove pegs in such a way that every strategic point has a peg at some step.

Yesterday it was raining, so the rock will be wet and it can be unsafe to remove pegs. Because of this, Fiona will only remove a peg $p$ if she can stand on the same pegs as when $p$ was placed. Alas Fiona’s existing plans do not take this new precaution into account, so Fiona has to update her plans and she has asked you for help. She would like not to carry too many extra pegs, so you promised to find safe plans using at most $10$ times more pegs than her previous plans were using. Can you deliver on your promise?

For example, consider the wall in the first sample input with $5$ strategic points. Point $1$ is close to the ground so it does not depend on any point. There has to be a peg in point $1$ in order to put a peg in point $2$, and the same holds for point $3$. In order to put a peg in point $4$, there has to be a peg both in point $2$ and point $3$. To put a peg in point $5$ it is enough if there is a peg at point $4$.

Therefore, the sequence (with annotations $+$ and $-$ depending on whether we add or remove a peg) $+1,+2,+3,-1,+4,-2,-3,+5$ is a safe dry plan, and it uses $3$ pegs. However it is not a safe wet plan, because we remove the pegs at points $2$ and $3$ without support. The sequence $+1,+2,-2,+3,-1,+4,-3,+5$ only requires $2$ pegs, but it is not safe at all because we add a peg to point $4$ without there being a peg at point $2$. The sequence $+1,+2,+3,-1,+4,+5$ is a safe wet plan, and it uses $4$ pegs.

-----Input-----
The first line contains an integer $n$ ($1 \leq n \leq 1000$), the number of strategic points in a wall. Each of the following $n$ lines contains an integer $p$ ($0 \leq p < n$) and a list of $p$ integers. If line $i$ contains $x_1,\ldots ,x_ p$ ($1 \leq x_ j < i$), then all points $x_ j$ need to have a peg in order to place a peg in point $i$.

The next line contains an integer $t$ ($1 \leq t \leq 1000$), the number of steps in the safe dry plan. Each of the following $t$ lines contains an integer $i$, meaning that a peg is being placed or removed from point $i$.

-----Output-----
If there is no safe wet plan using at most $10$ times the number of pegs of the safe dry plan, output $-1$. Otherwise, the first line must contain an integer $t$ ($1 \leq t \leq 1000000$), the number of steps in the safe wet plan. Each of the next $t$ lines must contain an integer $i$, meaning that a peg is being placed or removed from point $i$.

-----Examples-----
Sample Input:
5
0
1 1
1 1
2 2 3
1 4
8
1
2
3
1
4
2
3
5
Sample Output:
6
1
2
3
1
4
5
user: Ivan is planning a large European tour with his jazz band. There are a total of $n$ cities in Europe, numbered with integers $1$ through $n$. Ivan is planning $d$ concerts in cities $a_1, a_2, \dots , a_ d$ in that exact order, never having two consecutive concerts in the same city ($a_ i \not= a_{i+1}$), possibly visiting some of the cities many times and, finally, ending the tour in the same city where it begun ($a_1 = a_ d$).

Ivan always takes a direct flight between cities $a_ i$ and $a_{i+1}$. However, he is trying to be smart with his ticket purchases in order to save money. As you may know, airlines price tickets based on supply and demand and, for example, it may be possible that one-way tickets are more expensive than round trip tickets between same cities. Generally, there are two kinds of tickets available for purchase:
 - One-way ticket from the origin city $a$ to destination city $b$ can be used to fly from $a$ to $b$ once (but not in the opposite direction).
 - Round trip ticket from the origin city $a$ to destination city $b$ can be used to fly once from $a$ to $b$, and once from $b$ to $a$. The return segment (from $b$ to $a$) does not need to be used. However, the segments have to be flown in order – it is not allowed for Ivan to use the return segment of $a$ ticket to fly from $b$ to a unless he has used the first segment of that ticket to fly from $a$ to $b$ before.

You are given a list of available airfares, find the least amount of money Ivan needs to spend on tickets to be able to complete his journey. Ivan can purchase an arbitrary number of tickets for each airfare. Once again, Ivan needs to take a direct flight from $a_ i$ to $a_{i+1}$ for every $i = 1, 2, \dots , d - 1$. You may assume that is possible to complete the journey using the airfares.

-----Input-----
The first line contains two integers $n$ and $d$ ($2 \le n, d \le 300000$) – the number of cities in Europe and the number of concerts. The following line contains integers $a_1, a_2, \dots , a_ d$ ($1 \le ai \le n, ai \not= a_{i+1}, a_1 = a_ d$) – the planned tour schedule.

The following line contains an integer $m$ ($3 \le m \le 300000$) – the number of airfares. The $k$-th of the following $m$ lines contains four tokens $s_ k$, $d_ k$, $t_ k$, $p_ k$, describing the $k$-th airfare as follows:
 - $s_ k$ and $d_ k$ ($1 \le s_ k, d_ k \le n, s_ k \not= d_ k$) are the origin and the destination city respectively,
 - $t_ k$ is an uppercase letter “O” or “R” denoting a one-way or round trip ticket respectively,
 - $p_ k$ ($1 \le pk \le 10^9$) is the ticket price, an integer.

-----Output-----
Output the least amount of money necessary to purchase tickets that allow Ivan to complete the planned tour.

-----Examples-----
Sample Input:
2 5
1 2 1 2 1
4
1 2 R 6
1 2 O 3
2 1 O 3
1 2 R 5
Sample Output:
10
user: Topologically sorting the nodes of a directed graph (digraph) $G$ means putting the nodes in linear order\[ v_0, v_1, v_2, v_3, \ldots \]

such that whenever there is an edge from node $x$ to node $y$ in $G$, $x$ always precedes $y$ in the linear ordering. It is not hard to show that the nodes of a digraph can be topologically sorted if and only if the graph is acyclic (does not contain any directed cycles).

Kahn’s Algorithm is a well-known topological sorting algorithm. Beginning with a digraph $G$ and an empty list $L$, it repeatedly executes the following three steps:
 - Let $S$ be the set of source nodes, i.e., the nodes with no incoming edges. If $S$ is empty, terminate the algorithm.
 - Let $\alpha $ be any node in $S$. Remove $\alpha $ and all its outgoing edges from $G$. If this removal of edges creates new source nodes, add these to $S$.
 - Insert $\alpha $ at the end of $L$.

Once Kahn’s Algorithm terminates, if $G$ is now empty, then $L$ contains the nodes of the initial graph in topologically sorted order (such an ordering may not be unique). Otherwise, $G$ has one or more cycles, and therefore topological sorting is impossible.

Regardless of the outcome of Kahn’s Algorithm, consider any iteration of Step $1$ above. If $S$ contains more than one node, then $\alpha $ can be any one of them, and the choice of $\alpha $ affects the composition of $S$ in later iterations, when further choices may be made that affect the composition of $S$ in yet later iterations, and so on. Taking into account all possible sequences of choices of $\alpha $ during the execution of Kahn’s Algorithm, what is the largest $S$ can ever be at the beginning of Step $1$?

-----Input-----
The input specifies a single directed graph. The first line of input contains two space-separated integers, $n$ and $m$ ($1 \leq n \leq 500$,  $0 \leq m \leq n(n-1)$), where $n$ is the number of nodes and $m$ is the number of edges. This is followed by $m$ lines, each containing two space-separated integers, $x$ and $y$ ($0 \leq x,y \leq (n-1)$, $x \neq y$), indicating that there is a directed edge from node $x$ to node $y$, where nodes are indexed $0, 1, 2, \ldots , (n-1)$. No edge will be listed more than once.

-----Output-----
Output a line containing a single integer: the largest possible size of $S$ at the beginning of any iteration of Step $1$ in the execution of Kahn’s Algorithm.

-----Examples-----
Sample Input:
4 3
0 1
1 2
2 3
Sample Output:
1