meetforever/GRPO_CL huggingface dataset:
user: Rarity is designing new clothes, and she is looking for some inspiration! Recently, she learned about the importance of symmetry, and is trying to find ways to incorporate this idea into her designs.

To learn more about symmetry, she decided to search for inspiration in a naturally-symmetric concept, palindromes. A palindrome is a string that reads the same forwards and backwards. For example, peep and deified are palindromes, while tepee and defied are not.

A substring of a string is a string that can be formed by removing zero or more characters from the start and end of the other string. For example, sig and design are substrings of design, while dig and signed are not.

Rarity wants to find the perfect amount of symmetry to incorporate into her new design; to this end, she wants to find a string consisting only of lowercase Latin alphabet characters that
 - has length exactly $N$,
 - has exactly $K$ distinct characters, and
 - whose longest palindromic substring has length exactly $P$.

Help her find any string that satisfies all her requirements, or tell her that what she wants is impossible.

-----Input-----
The first and only line of input contains exactly three integers, $N$ ($1 \leq N \leq 10^6$), $K$ ($1 \leq K \leq 26$) and $P$ ($1 \leq P \leq N$), the requirements of the string.

-----Output-----
Output on a line by itself:
 - a string satisfying all her requirements, if any such string exists, or
 - IMPOSSIBLE, otherwise.

If there are multiple correct answers, you can output any of them.

-----Examples-----
Sample Input 1:
6 5 3
Sample Output 1:
rarity

Sample Input 2:
9 8 1
Sample Output 2:
canterlot

Sample Input 3:
5 3 5
Sample Output 3:
madam
user: Thore Husfeldt likes to compare his time on solved Kattis problems with other people. If he has written the fastest solution on a problem he screams “I’m amazing” in pure joy. However if his solution is not the fastest he likes to find out if he is the best Thore, or maybe the best ThoreH on the scoreboard. He is the best ThoreH if there is another Thore above him, but no ThoreH above him. Once a terrible incident took place where Thore misread the scoreboard and thought that he was the best, but instead it was some other Thore who was amazing. Now he needs your help to find the smallest prefix that uniquely determines himself on the scoreboard. 

-----Input-----
The first line of the input contains a single number $n$, where $1 < n \leq 100$.

Then follows $n$ lines with a name on each line.

Each name only contains characters from the set $\big [\text {a-zA-Z}\big ]$, i.e. lower and upper case English letters. Each name is at least one character and at most $100$ characters long.

Exactly one name is “ThoreHusfeldt” in the input.

-----Output-----
The output is a single line. If “ThoreHusfeldt” is first on the scoreboard, output “Thore is awesome”. If another person, with a name containing “ThoreHusfeld”(note: no trailing “t”) as a prefix is above Thore, output “Thore sucks”.

Otherwise output the smallest prefix that uniquely determines Thore on the scoreboard from the top. In other words: find the smallest prefix of “ThoreHusfeldt” that no-one above “ThoreHusfeldt” shares.

-----Sample Explanation-----
In Sample Input $1$ there is another Thore above “ThoreHusfeldt”. However, “ThoreHusfeldt” is still the best “ThoreH”.

-----Examples-----
Sample Input 1:
2
ThoreTiemann
ThoreHusfeldt
Sample Output 1:
ThoreH

Sample Input 2:
2
ThoreHusfeldt
JohanSannemo
Sample Output 2:
Thore is awesome

Sample Input 3:
2
ThoreHusfeldter
ThoreHusfeldt
Sample Output 3:
Thore sucks
user: In an infinite binary tree:
 - Each node has exactly two children – a left and a right child.
 - If a node is labeled with the integer $X$, then its left child is labeled $2X$ and its right child $2X+1$.
 - The root of the tree is labeled 1.

A walk on the binary tree starts in the root. Each step in the walk is either a jump onto the left child, onto the right child, or pause for rest (stay in the same node).

A walk is described with a string of letters ’L’, ’R’ and ’P’:
 - ’L’ represents a jump to the left child;
 - ’R’ represents a jump to the right child;
 - ’P’ represents a pause.

The value of the walk is the label of the node we end up on. For example, the value of the walk LR is 5, while the value of the walk RPP is 3.

A set of walks is described by a string of characters ’L’, ’R’, ’P’ and ’*’. Each ’*’ can be any of the three moves; the set of walks contains all walks matching the pattern.

For example, the set L*R contains the walks LLR, LRR and LPR. The set ** contains the walks LL, LR, LP, RL, RR, RP, PL, PR and PP.

Finally, the value of a set of walks is the sum of values of all walks in the set. Calculate the value of the given set of walks.

-----Input-----
A string describing the set. Only characters ’L’, ’R’, ’P’ and ’*’ will appear and there will be at most $10000$ of them.

-----Output-----
Output the value of the set.

-----Examples-----
Sample Input 1:
P*P
Sample Output 1:
6

Sample Input 2:
L*R
Sample Output 2:
25

Sample Input 3:
**
Sample Output 3:
33
user: According to Mårten’s Theorem, everything which rhymes is the same thing. For example, Sheep is the same thing as Heap, Dictionary Attack is the same thing as Stack, and Tree is the same thing as the Mediterranean Sea.

Strangely, some people have objected to Mårten’s Theorem, despite being rigorously proved by Johan’s infinitely many self-referencing recursive lemmas. You have been given the task of deciding if a number of statements are consistent under Mårten’s Theorem.

You will be given a number of statements of the form "$X$ is $Y$" or "$X$ not $Y$", where $X$ and $Y$ are words consisting solely of the characters ’a’-’z’, with a length of 1-20 characters, denoting that $X$ is or isn’t equivalent to $Y$. You must decide whether the statements given are consistent with each other. For example, the two statements ulf is lukas and ulf not lukas contradict each other, and are not consistent with each other.

However, you must also take Mårten’s theorem into account. For the sake of simplicity, we say that two words rhyme if the last $\min (3, |X|, |Y|)$ characters are the same in two words. So if this condition holds for two words $X$ and $Y$ that are in the list of statements, it is equivalent to having a statement "$X$ is $Y$".

Note that words which do not appear in any statement do not exist, and should not be considered rhyming with anything. That is, the two words foo and moo do not rhyme with each other, unless the word oo or o exists in the input.

-----Input-----
The first line consists of an integer $0 \le N \le 100000$, the number of statements. The next $N$ lines consists of statements of the two given forms.

-----Output-----
Output should consist of the string "yes", if the statements are consistent with each other, i.e. everything which is stated to be the same either explicitly or by Mårten’s Theorem, really is the same. If you can find contradiction, output the string "wait what?".

-----Examples-----
Sample Input 1:
4
herp is derp
derp is herp
herp is herp
derp is derp
Sample Output 1:
yes

Sample Input 2:
3
oskar not lukas
oskar is poptart
lukas is smart
Sample Output 2:
wait what?

Sample Input 3:
1
moo not foo
Sample Output 3:
yes
user: It is 2178, and alien life has been discovered on a distant planet. There seems to be only one species on the planet and they do not reproduce as animals on Earth do. Even more amazing, the genetic makeup of every single organism is identical!

The genetic makeup of each organism is a single sequence of nucleotides. The nucleotides come in three types, denoted by ‘A’ (Adenine), ‘C’ (Cytosine), and ‘M’ (Muamine). According to one hypothesis, evolution on this planet occurs when a new nucleotide is inserted somewhere into the genetic sequence of an existing organism. If this change is evolutionarily advantageous, then organisms with the new sequence quickly replace ones with the old sequence.

It was originally thought that the current species evolved this way from a single, very simple organism with a single-nucleotide genetic sequence, by way of mutations as described above. However, fossil evidence suggests that this might not have been the case. Right now, the research team you are working with is trying to validate the concept of “parallel evolution” – that there might actually have been two evolutionary paths evolving in the fashion described above, and eventually both paths evolved to the single species present on the planet today. Your task is to verify whether the parallel evolution hypothesis is consistent with the genetic material found in the fossil samples gathered by your team.

-----Input-----
The input begins with a number $n$ ($1\le n\le 4000$) denoting the number of nucleotide sequences found in the fossils. The second line describes the nucleotide sequence of the species currently living on the planet. Each of the next $n$ lines describes one nucleotide sequence found in the fossils.

Each nucleotide sequence consists of a string of at least one but no more than $4000$ letters. The strings contain only upper-case letters A, C, and M. All the nucleotide sequences, including that of the currently live species, are distinct.

-----Output-----
Display an example of how the nucleotide sequences in the fossil record participate in two evolutionary paths. The example should begin with one line containing two integers $s_1$ and $s_2$, the number of nucleotide sequences in the fossil record that participate in the first path and second path, respectively. This should be followed by $s_1$ lines containing the sequences attributed to the first path, in chronological order (from the earliest), and then $s_2$ lines containing the sequences attributed to the second path, also in chronological order. If there are multiple examples, display any one of them. If it is possible that a sequence could appear in the genetic history of both species, your example should assign it to exactly one of the evolutionary paths.

If it is impossible for all the fossil material to come from two evolutionary paths, display the word impossible.

-----Examples-----
Sample Input 1:
5
AACCMMAA
ACA
MM
ACMAA
AA
A
Sample Output 1:
1 4
MM
A
AA
ACA
ACMAA

Sample Input 2:
3
ACMA
ACM
ACA
AMA
Sample Output 2:
impossible

Sample Input 3:
1
AM
MA
Sample Output 3:
impossible
user: Gwen loves most numbers. In fact, she loves every number that is not a multiple of $n$ (she really hates the number $n$). For her friends’ birthdays this year, Gwen has decided to draw each of them a sequence of $n-1$ flowers. Each of the flowers will contain between $1$ and $n-1$ flower petals (inclusive). Because of her hatred of multiples of $n$, the total number of petals in any non-empty contiguous subsequence of flowers cannot be a multiple of $n$. For example, if $n = 5$, then the top two paintings are valid, while the bottom painting is not valid since the second, third and fourth flowers have a total of $10$ petals. (The top two images are Sample Input $3$ and $4$.)

Gwen wants her paintings to be unique, so no two paintings will have the same sequence of flowers. To keep track of this, Gwen recorded each painting as a sequence of $n-1$ numbers specifying the number of petals in each flower from left to right. She has written down all valid sequences of length $n-1$ in lexicographical order. A sequence $a_1,a_2,\dots , a_{n-1}$ is lexicographically smaller than $b_1, b_2, \dots , b_{n-1}$ if there exists an index $k$ such that $a_ i = b_ i$ for $i < k$ and $a_ k < b_ k$.

What is the $k$th sequence on Gwen’s list?

-----Input-----
The input consists of a single line containing two integers $n$ ($2 \leq n \leq 1000$), which is Gwen’s hated number, and $k$ ($1 \leq k \leq 10^{18}$), which is the index of the valid sequence in question if all valid sequences were ordered lexicographically. It is guaranteed that there exist at least $k$ valid sequences for this value of $n$.

-----Output-----
Display the $k$th sequence on Gwen’s list.

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

Sample Input 2:
2 1
Sample Output 2:
1

Sample Input 3:
5 22
Sample Output 3:
4 3 4 2
user: You are programming an autonomous rocket powered hovercraft. The vehicle can travel very, very fast, but turning is difficult. Since it’s hovering, it turns by firing directional thrusters, which will turn the vehicle whether it is moving forward or is stationary.

The salt flats on which you’re testing have been mapped with a 2D Cartesian grid. The hovercraft starts at location $(00)$ on this grid, facing in the positive $X$ direction. Your job is to get the vehicle to another location $(xy)$ on the flats.

The hovercraft has a fixed ground speed $v$ miles per second and a fixed rate of rotation $w$ radians per second. Due to the power of the rockets both for acceleration and deceleration, it can attain its maximum speed virtually instantaneously, and come to a stop from maximum speed just as quickly. Likewise, it can begin rotating at its fixed rate instantaneously, and stop just as quickly. It can rotate either clockwise or counter-clockwise.

You must figure out the minimum amount of time to get the vehicle to its target. The program which controls the vehicle can start forward motion, stop forward motion, start rotating, and stop rotating, each exactly once. Note that starting/stopping forward movement can be independent of starting/stopping rotation.

-----Input-----
Each test case will consist of exactly two lines.

The first line will contain two space-separated integers $x$ and $y$ ($-1000\! \le \! x,y\! \le \! 1000, (xy){\neq }(00)$), which indicate the location on the grid mapped onto the flats that you are trying to reach, in units of miles.

The second line of input will contain two space-separated real numbers with exactly two decimal places: $v$ ($0.01\! \le \! v\! \le \! 10.00$) and $w$ ($0.01\! \le \! w\! \le \! 10.00$), where $v$ is the fixed speed of travel in miles per second, and $w$ is the fixed rate of rotation in either direction in radians per second.

-----Output-----
Output a single real number, which is the minimum amount of time (in seconds) it would take get the hovercraft from $(00)$ to $(xy)$ subject to the constraints. Your answer will be accepted if it is within an absolute error of $10^{-3}$.

-----Examples-----
Sample Input 1:
20 0
1.00 0.10
Sample Output 1:
20.00000000

Sample Input 2:
-10 10
10.00 1.00
Sample Output 2:
3.14159265

Sample Input 3:
0 20
1.00 0.10
Sample Output 3:
28.26445910
user: In a little town called Križ live $N$ people. Each of them has borrowed some money from exactly one other inhabitant. Now the time has come to pay back all the debts, but the problem is that everybody has spent all of their money!

The major of Križ has decided to solve this problem. The town will give money to a few people so that they can pay back their debts. When some people get their money back, a chain reaction is started – for example: person $A$ gets money from the city. Person $A$ uses that money to pay the debt toward person $B$. Person $B$ then uses that money to pay the debt towards person $C$ etc. If person $B$ didn’t have enough money to pay back the debt, they wait until they get enough. If they have more than enough money, person $B$ will keep what is left after payback.

Another example: if two people live in Križ, and they owe $$100$ to each other, the town will give $$100$ to one of them so they can pay back the debt to the other one.

Your task is to calculate the minimum total amount of money the town has to give to some subset of the inhabitants so that after the payback protocol described above all debts are payed.

-----Input-----
First line of input contains one integer $N$ ($2 \le N \le 200000$), number of inhabitants of Križ. They are numbered from $1$ to $N$.

The following $N$ lines contain two integers, separated by space. In the $i$’th of those lines, the first number $A_ i$ represents the id of the person $i$’th person owes money to ($1 \le A_ i \le N$, $A_ i \ne i$), and the second number $B_ i$ represents the ammount of the debt in $ ($1 \le B_ i \le 10000$).

-----Output-----
The first and only line of output should contain one integer – the minimum total ammount of money town has to give to its inhabitants so all debts are returned.

-----Examples-----
Sample Input 1:
4
2 100
1 100
4 70
3 70
Sample Output 1:
170

Sample Input 2:
3
2 120
3 50
2 80
Sample Output 2:
150
user: A bicycle race is being organized in a country. The transport network of the country consists of $N$ cities numbered $1$ through $N$, with $M$ bidirectional roads connecting them. We will use the following terms:
 - A path is a sequence of roads in which each road starts in the city the preceding road ended in.
 - A simple path is a path which never visits a city more than once.
 - A ring is a simple path ending in the same city it started in.

The network is such that there is at least one path between every pair of cities. Additionally, every road in the network is part of at most one ring.

Your task is to find the longest path for the race satisfying two constraints:
 - The path may begin in any city, but must end in city $1$.
 - The path may visit a city more than once, but it must not contain any road more than once.

-----Input-----
The first line of input contains two integers $N$ and $M$ ($2 \le N \le 10000$, $1 \le M \le 2N-2$) – the numbers of cities and roads in the network.

Each of the following $M$ lines contains two different integers $A$ and $B$ ($1 \le A, B \le N$). These numbers indicate that there is a bidirectional road between cities $A$ and $B$. No two cities will be directly connected by more than one road.

-----Output-----
Output the length of the longest race path on a single line.

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

Sample Input 2:
6 6
1 2
1 3
2 4
3 4
3 5
5 6
Sample Output 2:
5
user: Being a space captain can be a dangerous endeavour, especially when conducting missions in Borg space. Today your captain has been tasked with beaming aboard a Borg spaceship to see if he can discover anything useful. The Borg ship consist of rooms connected with passages of equal length, and with no more than one direct passage between any two rooms. It is possible to find a path from any room to any other. 

On the Borg ship there is also a sentry, placed there by the Borg eons ago. This sentry operates according to a rather unsophisticated algorithm, namely walking at random. This gives the sentry one advantage, however: You will never know where it is! More precisely, once every minute, the sentry chooses one of the neighbouring rooms with uniform probability, and walks quickly to this room. Your captain will also be spending exactly one minute in each room, and to minimise the chances that he will meet the sentry, you time his movement such that he and the sentry move at exactly the same time. Thus he will be caught by the sentry if and only if the two of them move into the same room at the same time, or if they try to swap rooms.

Star Fleet has issued as a direct order the rooms your captain is to visit, and in the exact order. These rooms form a walk on the Borg ship, where each room may be visited several times. Star Fleet has also provided you with a map of the ship. The captain will be beamed down to the first room of the walk, and will be beamed back up from the last room. He risks capture in both these rooms.

Now the captain starts wondering about the chances of success on this mission, hoping them to be very low indeed. After all, that makes it all the more interesting! Getting curious yourself, you decide you want to figure this out. Unfortunately the ship’s android is experiencing problems with his new emotion chip, and thus the task falls on you to compute the captain’s chances of success on this dangerous mission!

-----Input-----
One line with $2 \leq N \leq 500$ – the number of nodes.

One line with $1 \leq L \leq 500$ – the number of rooms the captain must visit.

One line with $L$ integers describing the captain’s walk, i.e., they give the exact walk the captain must perform.

$N$ lines beginning with an integer $n_ i$ – the number of neighbours of node $i$ – followed by $n_ i$ integers – the neighbours of node $i$, $0$-indexed.

-----Output-----
The chance the captain will be able to complete his mission without being discovered by the Borg sentry. The answer should be correct within an absolute or relative error of $10^{-6}$

-----Examples-----
Sample Input 1:
3
1
0
2 1 2
1 0
1 0
Sample Output 1:
0.5

Sample Input 2:
8
6
1 0 2 3 0 1
7 1 2 3 4 5 6 7
1 0
2 0 3
2 0 2
1 0
1 0
1 0
1 0
Sample Output 2:
0.0446429
user: Raelynn is trying to learn the newest craze in modern art: Tape Art! This wonderful new type of art is created by taking a wooden plank and pieces of tape of different colors. Each artwork is constructed by taking multiple pieces of tape and placing them on the plank. For each color that appears in the artwork, only a single piece of tape is used. Tapes can be placed on top of each other in which case the tape placed last obscures already placed pieces of tape with which it overlaps. 

Raelynn has decided the best way to learn is by copying Sheila, the world-famous tape artist. Unfortunately those pieces of art are under lock and key and Raelynn can see only pictures of these marvels. Since Raelynn is having trouble reverse engineering the artwork from the picture, she has hired you to create a set of instructions with which she can copy the art.

Since Raelynn is spoiled by the ease of IKEA catalogs she requires instructions to be given in the following format: there should only be one instruction per color of tape and instructions should be given in the order they should be executed.

-----Input-----
The input consists of a single test case. The first line of this test case contains one integer $n$ ($1 \le n \le 10^5$), where $n$ is the length of the tape art in inches. The next line contains $n$ integers $c_ i$ ($1 \le c_ i \le n$) representing the color of one inch of the plank. Planks are divided into $n$ $1$-inch sections numbered $1$ through $n$.

-----Output-----
Output any set of instructions that, when executed, will result in the tape art given by the input. The set of instructions should start with the number of instructions that follow. Each instruction must consist of three numbers: $l$$r$ $c$ where $[l, r]$ represents the inclusive range on which the tape should be placed and $c$ represents the color of the tape piece.

Output the string “IMPOSSIBLE” if the piece of tape art cannot be reconstructed using only one piece of each color (Sheila must have broken the rules to make it or this piece is a forgery).

-----Examples-----
Sample Input 1:
6
1 2 3 3 2 1
Sample Output 1:
3
1 6 1
2 5 2
3 4 3

Sample Input 2:
4
1 2 1 2
Sample Output 2:
IMPOSSIBLE