meetforever/GRPO_CL huggingface dataset:
user: At art shows, it is very common to have booths where children can create their very own sand art. This art is typically made by taking a jar or bottle and filling it with layers of different colors of sand. Instead of a bottle, this year a new container is being used for decorating! The container is a glass box!

The box has a 2D rectangular face and a thickness of exactly 1 unit. Inside the glass box, $n-1$ vertical dividers are placed to separate it into $n$ sections. In the example below, the box is divided into 4 sections using 3 dividers:

Sometimes the children want certain amounts of each color to be in each section of their work of art. They specify a minimum and maximum for each combination of section and sand color. Your task is to help them find how balanced they can make the artwork. This is done by minimizing the difference between the sand heights in the section with the highest overall sand level and the section with the lowest overall sand level.

-----Input-----
Each input will consist of a single test case. Note that your program may be run multiple times on different inputs. Each input begins with a single line with 4 space-separated integers, $n\ m\ w\ h$, where:
 - $n$ ($2 \le n \le 200$) is the number of sections
 - $m$ ($1 \le m \le 200$) is the number of colors of sand
 - $w,h$ ($1 \le w,h \le 5000$) are the width and height of the box (it always has a depth of 1)

The next line will contain $m$ space-separated real numbers (with at most 3 decimal places) $v$ ($0<v\le w\cdot h$), which represent the volumes of each color of sand. It is not necessary to use all of the sand, but the minimums for each section must be satisfied.

The next line will have $n-1$ space-separated real numbers with at most 3 decimal places) $x$ ($0< x < w$) which represent the distance from the left wall of each divider. The $x$s are guaranteed to be sorted in increasing order.

The next $n$ lines will each have $m$ space-separated real numbers (with at most 3 decimal places) $min$ ($0 \le min \le w\cdot h$). The $j$th element of the $i$th row is the minimum amount of sand color $j$ to be put in section $i$.

The next $n$ lines will each have $m$ space-separated real numbers (with at most 3 decimal places) $max$ ($0 \le max \le w\cdot h$). The $j$th element of the $i$th row is the maximum amount of sand color $j$ to be put in section $i$, and $min_{ij} \le max_{ij}$.

-----Output-----
Output a real number rounded to exactly 3 decimal places representing the minimum difference possible between the maximum and minimum heights of sand in the sections. A distribution of sand will always be possible that satisfies the constraints in the input.

-----Examples-----
Sample Input 1:
2 2 5 5
2.0 2.0
4.0
1.0 0.0
0.0 1.0
1.0 0.0
0.0 2.0
Sample Output 1:
0.750

Sample Input 2:
2 2 5 5
2.0 2.0
4.0
1.0 0.0
0.0 1.0
1.5 0.0
0.0 2.0
Sample Output 2:
0.625
user: You’re one of the king’s spies sent on a secret mission to retrieve an item of incredible value, an ancient scroll from the throne room. Legend has it, the scroll contains the answer to the $P$ versus $NP$ problem. When you finally reach the throne room, you realize there is a code the guards enter every day while observing them. As a spy, you’ve started to notice a few rules for each guard’s code:
 - each code is a matrix consisting of nonzero decimal digits (integers from $1$ to $9$) with $m$ rows and $n$ columns
 - no digit repeats within any row of the code
 - for each digit $l$ in the code, except for those in the topmost row and rightmost column, let $u$ be the digit above it and let $r$ be the digit to its right in the code matrix. Then one of the following must be true:
 - $u$ is the product of $l$ and $r$
 - $u$ is the sum of $l$ and $r$
 - $u$ is the difference of $l$ and $r$ or $r$ and $l$
 - $u$ is the quotient of $l$ and $r$ or $r$ and $l$

On day $999$, you’ve noticed a guard has seem to walked off while entering his code. Some digits have been omitted, but after careful consideration you think you can crack the code. Digits that have been omitted are represented with a $0$. How many complete codes are possible, given the guard’s partial code?

-----Input-----
A test case starts with a line containing two numbers $m$ ($3 \le m \le 6$) and $n$ ($3 \le n \le 6$), which is the number of rows and number of columns of the grid. The following $m$ lines contain $n$ integers from $0$ to $9$, separated by spaces. $0$ indicates an unknown value that you can supply, and there will be at most $\lfloor \frac{m*n}{2} \rfloor $ unknown values.

You can assume the guard has followed the rules with the partial code (i.e. no repeated digits appear in any row in the input, and any three pairs of non-zero digits that form an L have the property described above).

-----Output-----
For each test case, print the number of complete codes you can find.

-----Examples-----
Sample Input 1:
3 3
1 2 4
0 3 6
4 0 3
Sample Output 1:
2

Sample Input 2:
3 4
2 3 0 7
0 0 2 1
0 0 3 0
Sample Output 2:
37
user: Luka is not paying attention in class again, while the teacher is explaining redox reactions. Instead of paying attention, he is playing with analog dials.

An analog dial is a small device which always shows one digit between $0$ and $9$. It also contains a small button which increases the number by $1$ (unless it is $9$, in which case it is changed to $0$).

Luka has $N$ such dials on his desk, numbered $1$ to $N$ left to right, and two sheets of paper for him to write on.

Luka’s game starts with him setting the dials in some starting configuration, which he then writes onto the first sheet. Luka then does the following $M$ times:
 - Choose two integers $A$ and $B$ ($1 \le A \le B \le N$) and write them down on the first sheet.
 - Calculate the sum of numbers on dials between $A$ and $B$ (inclusive), and write the sum down on the second sheet.
 - Press the button once on all dials between $A$ and $B$.

Just as he had finished his game, the teacher noticed him, and took away all his dials and the second sheet of paper.

Given the contents of the first sheet, help him calculate the numbers on the second sheet.

-----Input-----
The first line contains two integers $N$ and $M$ ($1 \le N \le 250000$, $1 \le M \le 100000$).

The second line contains the initial configuration of the dials, $N$ digits with no spaces. The first digit is the number initially on dial $1$, the second digit the number on dial $2$ and so on.

Each of the following $M$ lines contains two integers $A$ and $B$ ($1 \le A \le B \le N$).

-----Output-----
Output $M$ lines, the sums calculated by Luka, in order in which he calculated them.

-----Examples-----
Sample Input 1:
4 3
1234
1 4
1 4
1 4
Sample Output 1:
10
14
18

Sample Input 2:
4 4
1234
1 1
1 2
1 3
1 4
Sample Output 2:
1
4
9
16
user: Garry is looking at the sky. Such a beautiful day! He notices that the clouds are particularly beautiful today, and wishes to record the current state of the sky. He has no camera, so he begins writing down coordinate points. Fortunately for Garry, the current cloud cover can be represented as the union of non-intersecting, non-degenerate triangles where each vertex is at a coordinate point on the $xy$-plane. Two triangles are considered non-intersecting if their intersection has area $0$. A triangle is degenerate if its area is $0$.

The next day, Garry’s friend Jerry goes to look at the sky. Jerry also wishes to record the current state of the sky. He follows the same protocol as Garry, and writes down the cloud cover as a set of non-intersecting triangles.

Garry and Jerry want to determine if they saw the same cloud cover. Unfortunately, there are multiple possible ways to represent the same cloud cover! Given Garry and Jerry’s notes, did they see the same cloud cover in the sky?

-----Input-----
The first line of input contains the integer $n$, $(0 \le n \le 100000)$, the number of triangles Garry wrote down. Each of the next $n$ lines contains $6$ space separated integers, $x_1$, $y_1$, $x_2$, $y_2$, $x_3$, and $y_3$. These are Garry’s triangles. The next line contains the integer $m$, $(0 \le m \le 100000)$, the number of triangles Jerry wrote down. Each of the next $m$ lines contains $6$ space separated integers, $x_1$, $y_1$, $x_2$, $y_2$, $x_3$, and $y_3$. These are Jerry’s triangles. The absolute value of the $x$ and $y$ coordinates are at most $10^9$. (That’s as far as Garry and Jerry can see.)

-----Output-----
Print “yes” if Garry and Jerry saw the same cloud cover, or “no” if they did not.

-----Examples-----
Sample Input 1:
1
10000 0 10000 10000 0 10000
3
10000 0 10000 10000 5000 5000
5000 5000 10000 10000 0 10000
0 0 0 1000 1000 0
Sample Output 1:
no

Sample Input 2:
2
9996 0 9997 0 0 1
9999 0 10000 0 0 1
2
9997 0 9998 0 0 1
9998 0 9999 0 0 1
Sample Output 2:
no
user: In the ACM kingdom, there are $N$ cities connected by $M$ two-way roads. These cities are connected, i.e., one can reach from any city $X$ to any other city $Y$ by going through some of these roads. One day, the government wishes to assign for each road a direction, such that one can still reach from any city to any other. You are asked to determine whether this task is possible.

-----Input-----
The first line of each test case consists of two integers, $N$ ($1 \leq N \leq 50$), and $M$ ($1 \leq M \leq N(N - 1)/2$). Each of the next $M$ lines describes a road, and consists of two integers, $X$ and $Y$, ($1 \leq X, Y \leq N$; $X \neq Y$), indicating that there is a road between city $X$ and $Y$. There is at most one road that directly connects each pair of cities.

-----Output-----
If it is impossible, output a single line NO. Otherwise, output YES on the first line, followed by $M$ lines describing one possible direction assignment to these $M$ roads. Each of these $M$ lines should consist of two integers, $X$, $Y$, indicating that there is a one-way road from city $X$ to city $Y$. These $M$ lines can be output in any order.

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

Sample Input 2:
4 3
1 2
1 3
1 4
Sample Output 2:
NO
user: Your friend is an organizer of the International Chess Playing Championship. He is worried that some of the contestants may be cheating, and he has asked you to help out. The chess players are allowed to report matches to the jury themselves, and this is not checked with the reported opponent. So, it is possible for competitors to make up matches and falsely report themselves as the winners.

Since chess is a game of skill, and not of chance, a player will always beat their opponent if their skill level is higher. A game will result in a draw if and only if the two players’ skills are exactly equal.

However, the skill level of the players is not known. He has therefore asked you to write a program that, given a list of reported matches, determines whether this list is consistent or not. The list is inconsistent if we can determine that at least one reported match is falsely reported, otherwise it is consistent.

-----Input-----
The first line contains two integers $N$ ($2 \leq N \leq 50000$) and $M$ ($1 \leq M \leq 250000$), to describe a championship with $N$ players and $M$ reported matches.

The following $M$ lines each consist of an integer $K$, a symbol which is either ‘=’ or ‘>’, and another integer $L$. The integers $K$ and $L$ each uniquely identify a player ($0 \leq K, L < N$). If the symbol is ‘=’, then the game between $K$ and $L$ was a draw. If the symbol is ‘>’, then $K$ beat $L$ in a match.

You may assume that there is at most one reported match between any given pair of players. Also, each player takes part in at least one reported match.

-----Output-----
Output a single line containing a single word: “consistent” if the list of recorded matches is consistent, and “inconsistent” if it is not.

-----Examples-----
Sample Input 1:
3 3
0 > 1
1 = 2
0 = 2
Sample Output 1:
inconsistent

Sample Input 2:
5 5
0 = 1
1 = 2
3 = 4
0 > 3
1 > 4
Sample Output 2:
consistent
user: Vera is an astronomer studying locality of nearby star systems. The star systems can be thought of as 3D points in space. Vera would like to place a can around the stars. In other words, she would like to know what is the smallest volume cylinder that encloses the stars. The cylinder can be oriented in any direction. At least one base of the cylinder must have at least three stars.

-----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 $n$ ($4 \le n \le 1000$), representing the number of stars.

Each of the next $n$ lines will contain three integers $x$, $y$ and $z$ ($-1000 \le x, y, z \le 1000$), representing the position of a single star. No two stars will be at the same position. No four stars will be coplanar.

-----Output-----
Output a floating point number representing the smallest volume cylinder that can enclose all the stars. Your answer must be accurate within a relative tolerance of $10^{-6}$.

-----Examples-----
Sample Input 1:
4
1 0 0
1 1 0
0 0 0
0 0 1
Sample Output 1:
1.57079633

Sample Input 2:
4
-100 0 0
10 0 10
-10 -10 -10
0 0 0
Sample Output 2:
41938.65135885
user: Ivana won the bet (Zvonko hadn’t foreseen this and suspects that it is due to outside interference) and now Zvonko is waiting for her at the movies. While he is waiting, he is observing messages on a screen above him.

As Ivana is running late, Zvonko has been looking at the screen for a while and noticed that some messages appeared on the screen more than once. Naturally, he’s been writing down all messages on a piece of paper. He wants to know the length of the longest string that appeared at least twice (appears in two different positions on the paper).

-----Input-----
The first line of input contains an integer $L~ (1 \leq L \leq 200000)$, the length of the string Zvonko wrote down.

The second line contains a string of $L$ lowercase letter of the English alphabet.

-----Output-----
Output the length of the longest string that appears twice on a single line. If there is no such string, output zero.

-----Examples-----
Sample Input 1:
11
sabcabcfabc
Sample Output 1:
3

Sample Input 2:
18
trutrutiktiktappop
Sample Output 2:
4
user: Alice and Bob love playing Don’tminion, which typically involves a lot of shuffling of decks of different sizes. Because they play so often, they are not only very quick at shuffling, but also very consistent. Each time Alice shuffles her deck, her cards get permuted in the same way, just like Bob always permutes his cards the same way when he shuffles them. This isn’t good for playing games, but raises an interesting question.

They know that if they take turns shuffling, then at some point the deck will end up ordered in the same way as when they started. Alice shuffles once first, then Bob shuffles once, then Alice shuffles again, et cetera. They start with a sorted deck. What they do not know, however, is how many shuffles it will take before the deck is sorted again.

Can you help them compute how many shuffles it will take? As Alice and Bob can only do $10^{12}$ shuffles in the limited time they have, any number strictly larger than this should be returned as huge instead.

-----Input-----
 - The first line contains a single integer $1\leq n\leq 10^5$, the number of cards in the deck.
 - The second line contains $n$ distinct integers $1\leq a_1, a_2, \dotsc , a_ n \leq n$, where $a_ i$ is the new position of the card previously at position $i$ when Alice shuffles the deck.
 - The third line contains $n$ distinct integers $1\leq b_1,b_2,\dotsc , b_ n\leq n$, where $b_ i$ is the new position of the card previously at position $i$ when Bob shuffles the deck.

-----Output-----
 - Output a single positive integer $m > 0$, the minimal number of shuffles required to sort the deck, or huge when this number is strictly larger than $10^{12}$.

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

Sample Input 2:
6
5 1 6 3 2 4
4 6 5 1 3 2
Sample Output 2:
5
user: In programming contest circles, one of the most important roles is that of the Chief Equality Officer (CEO). This person is responsible for making sure that every team has an equal chance of winning the contest. Since last year’s NWERC the current CEO, Gregor, has thought at length about how to make the contest even more fair and equal.

His answer is to introduce a new programming language as the only one allowed for submissions. This way, no team will be disadvantaged by not having mastered any of the allowed languages. This language is called Balloon, short for Building A Long List Of Ordinary Numbers. Its only data type is the list of integers. To keep the language fast, it contains only four instructions:
 - [x$_1$,…,x$_ n$] is the constructor for lists. It returns the integers inside the brackets in their given order.
 - concat(<Expr$_1$>,<Expr$_2$>) returns a list of all the integers returned when evaluating the expression <Expr$_1$> followed by all of the integers returned when evaluating <Expr$_2$>.
 - shuffle(<Expr>) returns a list of all the integers returned by <Expr>, rearranged according to a uniformly random permutation, i.e., each permutation of the elements is used with equal probability.
 - sorted(<Expr>) returns a list of all the integers returned by <Expr>, rearranged into non-decreasing order.

As an example, consider the first expression of Sample Input 1. The two shuffle exressions both take the list [1,2] as input and return one of the lists [1,2] and [2,1], each with probability $0.5$ (independently of each other). The outer concat operator takes the two returned lists as its input and returns their concatenation. I.e., it returns one of the lists [1,2,1,2], [1,2,2,1], [2,1,1,2], and [2,1,2,1], each with probability $0.25$.

Naturally, we cannot use byte-by-byte output comparison any more when teams submit their solutions in Balloon, as its output is probabilistic. The judge server instead has to check whether a submitted program is equivalent to the sample solution created by the judges. Two programs are equivalent if for any list $L$ of integers, both programs have an equal probability of returning $L$.

It is your task to determine whether two given Balloon programs are equivalent.

-----Input-----
The input consists of:
 - One line containing a string A, the first program.
 - One line containing a string B, the second program.

Each program is a syntactically valid Balloon program with between $3$ and $10^6$ characters, and contains neither spacing nor empty lists (i.e., the strings “ ” or “[]” do not occur in the input).

Each integer in each program is greater than $0$ and less than $10^{9}$.

-----Output-----
If the two programs are equivalent, output “equal”, otherwise output “not equal”.

-----Examples-----
Sample Input 1:
concat(shuffle([1,2]),shuffle([1,2]))
shuffle([1,2,1,2])
Sample Output 1:
not equal

Sample Input 2:
sorted(concat([3,2,1],[4,5,6]))
[1,2,3,4,5,6]
Sample Output 2:
equal
user: The other day Unnar was playing the video game Geezer Scripts, Geezer Scripts V to be precise. He opened a door (in the game) and before he knew he was in a large and complicated cave-system, full of enemies who will attack as soon as Unnar gets to close to them. What’s worse is that the game doesn’t remember which enemies the player has already defeated so if the player leaves and comes back again the enemy will have respawned. Since Unnar is quite prone to getting lost he needs your help to find a way through the cave which incurs minimal damage to him.

Unnar’s character has some $A$ attack points and $H$ health points. The same is true for his enemies. When Unnar fights an enemy Unnar goes first. He subtracts his attack points from the enemies health points and the enemy then does the same. This process is repeated until either Unnar or the enemy has fewer than $1$ health points. The loser is the one who ran out of health points. If Unnar loses it’s Game Over, but if Unnar wins he can continue traversing the cave-system but his lost health points do not regenerate.

The cave-system is not evenly elevated so its passages are one-way. The cave-system can be modelled using $n$ areas connected with $m$ one-way passages. Unnar starts in area $1$ and wants to get to area $n$. Every passage contains an enemy but the areas are safe.

-----Input-----
The first line of the input contains two integers $1 \leq A, H \leq 10^9$ where $A$ denotes Unnar’s attack points and $H$ his health points. The second line contains $1 \leq n, m \leq 10^5$ where $n$ is the number of ares in the cave and $m$ the number of passages. The next $m$ lines contain four integers $1 \leq e_ i, b_ i \leq n$ and $1 \leq a_ i, h_ i \leq 10^9$, indicating a passage from area $e_ i$ to $b_ i$ whose enemy has $a_ i$ attack points and $h_ i$ health points.

-----Output-----
If Unnar can’t get through the cave-system output ‘Oh no’, without the quotes. Otherwise output the maximum health Unnar can have after getting through the cave-system.

-----Examples-----
Sample Input 1:
1 2
3 2
1 2 1 2
2 3 1 2
Sample Output 1:
Oh no

Sample Input 2:
1 3
3 2
1 2 1 2
2 3 1 2
Sample Output 2:
1
user: Amanda Air has routes between many different airports, and has asked their most important frequent flyers, members of the AA Frequent Flyer program, which routes they most often fly. Based on this survey, Amanda, the CEO and owner, has concluded that Amanda Air will place lounges at some of the airports at which they operate.

However, since there are so many routes going between a wide variety of airports, she has hired you to determine how many lounges she needs to build, if at all possible, given the constraints set by her. This calculation is to be provided by you, before any lounges are built. Her requirements specifies that for some routes, there must be lounges at both airports, for other routes, there must be lounges at exactly one of the airports, and for some routes, there will be no lounges at the airports.

She is very economically minded and is demanding the absolute minimum number of lounges to be built.

-----Input-----
The first line contains two non-negative integers $1 \leq n,m \leq 200\; 000$, giving the number of airports and routes in the Amanda Catalog respectively. Thereafter follow $m$ lines, each describing a route by three non-negative integers $1 \leq a, b \leq n$ and $c \in \{ 0,1,2\} $, where $a$ and $b$ are the airports the route connects and $c$ is the number of lounges.

No route connects any airport with itself, and for any two airports at most one requirement for that route is given. As one would expect, $0$ is a request for no lounge, $1$ for a lounge at exactly one of the two airports and $2$ for lounges at both airports.

-----Output-----
If it is possible to satisfy the requirements, give the minimum number of lounges necessary to do so. If it is not possible, output impossible.

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

Sample Input 2:
5 5
1 2 1
2 3 1
2 4 1
2 5 1
4 5 1
Sample Output 2:
impossible