meetforever/GRPO_CL huggingface dataset:
user: Yoshi is a frog. He lives in the ZOO under a log which was specially transported there from a distant equatorial rainforest. The log is big and wet and attracts the flies and that is what makes Yoshi satisfied.

There is also a line of pebbles which runs through the wetland in front of the log. There are various dark spots on the pebbles and sometimes Yoshi likes to look at them and dream that they are not just spots but some really big flies instead.

Yesterday, Yoshi’s friend camel Addawser came to see him and suggested to play a game.

“Do you see those spots on the pebbles?” asked Addawser. “I challenge you to start on the leftmost pebble and then do some jumps from a pebble to another pebble but with some restrictions. You can jump from a pebble to another one only if the sum of numbers of spots on both pebbles is equal to the distance between the pebbles. And you have to get to as distant pebble as possible.”

“All right, but you know that I can count at most to twenty three and no more,” hesitated Yoshi.

“No problem, I will help you with the bigger numbers,” said Addawser.

“Then, it’s easy,” said Yoshi, positioning himself on the first pebble and looking inquisitively over the rest of the line. “Although, one might not be quite so sure, after all,” he added after a while.

You are given the sequence of numbers of dark spots on the pebbles in the line. You are asked to find the most distant pebble which can be reached by a sequence of jumps. The first jump starts at the first pebble in the line and a jump between two pebbles is possible if and only if the sum of numbers of spots on both pebbles is equal to the distance between them. You may suppose that the line of pebbles is straight and that the distance between each two neighboring pebbles is exactly one frog distance unit.

-----Input-----
Each case starts with a line containing one integer $N$ ($1 \leq N \leq 1000000$) representing the number of pebbles. The second line contains a list of $N$ integers. The order of the integers in the list is the same as the order of the pebbles in the wetland, each integer represents the number of spots on the corresponding pebble. No pebble contains more than $10^9$ spots. Suppose that Addawser knows all different pairs of pebbles where Yoshi can perform a jump from one pebble to another one during his sequence of jumps. You are guaranteed that the number of those pairs of pebbles never exceeds $1000000$.

-----Output-----
Print a single line with one integer denoting the distance of the pebble which can be reached by successive jumps according to the given rules and which is the most distant from the first pebble.

-----Examples-----
Sample Input:
7
2 1 0 1 2 3 3
Sample Output:
5
user: You are hired as a level designer for the next Pokémon series, with games called Ice and Fire. For the first of these two games, players have to get through a maze in an icy cave. The cave is represented as a grid, with each square of the grid being either ice, gravel or an obstacle.

The player will start at a square, and then make a number of moves, each move represented by one of the four cardinal directions. The maze behaves in the following way. Assume that the square the player is trying to move into is an obstacle. In this case, the player does not move. If the square the player is moving into is gravel, the player successfully moves to the square and will stand still on the square. If the square is ice however, the player will first be transferred into that square, and then repeat the procedure again in the same direction. This means the player will glide on the ice until either colliding with an obstacle or reaching a square filled with gravel. Gliding on ice counts only as one move.

You have almost finished your level design. In the maze, there is a goal square that you wish to reach. You still have to choose a square to be the starting point of the player. Since you do not want the level to be too easy, you want to make sure the number of moves needed to get from the starting point to the goal is sufficiently high.

Can you compute the minimum number of moves needed to get from each point in the maze to the goal? Note that move may result in the player traveling multiple squares if gliding on the ice.

-----Input-----
The first line of the input contains the two integers $3 \le C \le 1000$ and $3 \le R \le 1000$, the number of columns and rows that the maze consists of.

The next $R$ lines contains $C$ characters each, describing the maze. Each square in the maze is represented by one of the following characters:
 - a period (.) represents a gravel square
 - a pound sign (#) represents an obstacle
 - an underscore (_) represents an ice square
 - an M (M) represents the goal in the maze, which is also covered in gravel

The edges of the maze are always surrounded by obstacle squares.

-----Output-----
Output $R$ lines with $C$ integers each, one for each square, containing the number of moves needed to reach the goal.

If it is not possible to reach the target from a square, output $-1$ instead for that square.

-----Examples-----
Sample Input:
5 6
#####
#...#
#_###
#_M.#
#__.#
#####
Sample Output:
-1 -1 -1 -1 -1
-1 4 5 6 -1
-1 4 -1 -1 -1
-1 1 0 1 -1
-1 3 1 2 -1
-1 -1 -1 -1 -1
user: The mayor of RMRCity wants to create a secure landline telephone network for emergency use in case of serious disasters when the city is cut off from the outside world. Some pairs of buildings in the city can be directly connected with a wire telephone line and the municipality engineers have prepared an estimate of the cost of connecting any such pair.

The mayor needs your help to find the cheapest network that connects all buildings in the city and satisfies a particular security measure that will be explained shortly. A call from a building $A$ to another building $B$ may be routed through any simple path in the network (i.e., a path that does not have any repeated building). There are also some insecure buildings that one or more persons with serious criminal records live in. The mayor wants only communications intended for these insecure buildings to reach them. In other words, no communication from any building $A$ to any building $B$ should pass through any insecure building $C$ in the network (where $C$ is different from $A$ and $B$).

-----Input-----
The input consists of a single test case. The first line contains three integers $n$, $m$, $p$ where $1 \le n \le 1000$ is the number of buildings, $0 \le m \le 100000$ is the number of possible direct connections between a pair of buildings, and $0 \le p \le n$ is the number of insecure buildings. The buildings are numbered from $1$ to $n$. The second line contains $p$ distinct integers between $1$ and $n$ (inclusive), which are the numbers of insecure buildings. Each of the next $m$ lines contains three integers $x_ i$, $y_ i$, and $\ell _ i$ describing one potential direct line, where $x_ i$ and $y_ i$ $(1 \le x_ i, y_ i \le n)$ are the distinct buildings the line connects, and $\ell _ i$ $(1 \le \ell _ i \le 10000)$ is the estimate of the cost of connecting these buildings. There is at most one direct link between any two buildings in these $m$ lines.

-----Output-----
Display the cost of the cheapest network satisfying the security measure if it is possible. Otherwise, display impossible.

-----Examples-----
Sample Input:
4 6 1
1
1 2 1
1 3 1
1 4 1
2 3 2
2 4 4
3 4 3
Sample Output:
6
user: Recall that a subsequence of a string is any string obtained by removing some subset of characters from the string, for instance “string”, “sing”, “i” and “sg” are all subsequences of “string”. If the same subsequence can be obtained in exactly $t$ different ways, by removing different subsets of characters, we say that the subsequence occurs $t$ times. 

Jingfei wants to create a nonempty bit string that has the following properties:
 - the subsequence 00 occurs $a$ times,
 - the subsequence 01 occurs $b$ times,
 - the subsequence 10 occurs $c$ times, and
 - the subsequence 11 occurs $d$ times.

However, Jingfei does not know how to create such a string – or whether it is even possible. Could you help her?

-----Input-----
The input consists of a single line with four integers $a$, $b$, $c$, and $d$ ($0 \leq a,b,c,d \leq 10^9$).

-----Output-----
Output a bit string that satisfies the given requirements. If there are several solutions, output any one of them. If there are no solutions, output “impossible”.

-----Examples-----
Sample Input:
3 4 2 1
Sample Output:
01001
user: Sector quadrant MAPS-2019 has been the home of the Rebel Alliance ever since the beginning of the space war against the Galactic Empire. Now news has reached the Rebel Council that the Empire is conspiring to attack the Rebel planets in the quadrant. Unfortunately, existing communication technology does not enable a distress signal to reach all the planets in time to warn them of the upcoming invasion. Catherine has been appointed by the Council to pilot her spaceship to each Rebel planet and deliver the warning, and then return to her home planet.

Luckily for Catherine, the Rebel Alliance has been making progress towards developing portal technology that allows instant transportation between any two planets! Every Rebel planet is equipped with exactly one portal in case of an emergency such as this one. A spaceship that enters the portal on one planet immediately exits the portal on another planet (whichever planet the pilot chooses). However, there is still one glitch the Rebel scientists have not yet figured out: either entering or exiting a portal causes it to collapse on itself, so each portal can only be used once.

A planet is small enough relative to the size of the galaxy that it can be represented as a single point in 3D space. The distance between any two planets is measured as the Euclidean distance between their representative points. To move from a planet $A$ to another planet $B$, Catherine has two choices: she can fly her spaceship in the usual way, travelling the Euclidean distance between $A$ and $B$, or she can enter the portal on $A$ and exit the portal on $B$ (assuming both portals still exist), which counts as a distance of $0$. Given the position of each planet, and the fact that each portal can only be used once, can you help Catherine plan her journey to deliver the warning to each Rebel planet and return to her home planet while travelling the shortest distance possible? Note that Catherine is allowed to revisit planets.

-----Input-----
The first line of input contains an integer $n$ $(2 \leq n \leq 18)$, the number of Rebel planets. This is followed by $n$ lines, each giving the position of one Rebel planet in 3D space, the first of which is Catherine’s home planet (where she begins her mission). Every planet position is a unique triplet of space-separated integers $x$ $y$ $z$ $(0 \leq x, y, z \leq 500)$.

-----Output-----
Output a single line containing the minimum total distance Catherine needs to travel in order to visit every Rebel planet and return to her home planet. Your answer will be accepted if it is within $10^{-6}$ of the official answer.

-----Examples-----
Sample Input:
4
0 0 1
0 1 1
2 0 3
2 1 3
Sample Output:
2.0
user: Luka is slacking again during chemistry class, while the teacher is explaining Avogadro’s law.

Luka first drew a table consisting of $3$ rows and $N$ columns. Then he wrote the numbers $1$ to $N$ into the first row in arbitrary order, each number appearing exactly once. In the other two rows he also wrote integers between $1$ and $N$, but didn’t care how many times a number appeared.

Luka can now delete any set of columns from the table. After doing so, he sorts the numbers in each row in ascending order.

He wants to obtain a table in which all three rows are identical after sorting. Write a program that determines the smallest number of columns he must delete.

For instance, in the first sample input below, Luka can delete the second, fourth, sixth and seventh columns. After deleting these and sorting each row, all three rows contain the numbers 1, 3 and 5.

-----Input-----
The first line of input contains the integer $N$ ($1 \le N \le 100000$), the number of columns in the table. The following three lines contain $N$ integers each, separated by single spaces. The numbers will be between $1$ and $N$, and there will be no duplicates in the first row.

-----Output-----
Output the smallest number of columns Luka must delete.

-----Examples-----
Sample Input:
7
5 4 3 2 1 6 7
5 5 1 1 3 4 7
3 7 1 4 5 6 2
Sample Output:
4
user: Tom the Straight Builder has finally finished building his Great Wall. Being a Straight Builder he has of course built a wall with no bends in it at all. And as far as you can see he has built this wall infinitely long and infinitely thin.

On one side of the wall Tom’s wife Ellen has decided she wants to make a perfect lawn. It is not very important to Ellen (or to Tom for that matter) what shape it has or exactly where it is located, but Ellen requires the lawn to be at least $L$ square meters.

To guard this pristine lawn from annoying trespassers Ellen has brought her dog William. William is a vicious dog and will attack any human being except Ellen upon sight. William can not cross the wall in any way. To reduce his own mutilations Tom has decided to tie William up with a chain to a pole and minimize the area he can cover while still being able to cover the whole lawn. The pole has been placed at location $(0,0)$ and Tom is not able to move it (it is after all the center of the universe).

Tom needs to know what length of dog chain he needs to buy in the hardware store. Specifically he wants to know the shortest length chain in whole meters he can buy so that the dog is able to cover an area of at least $L$ square meters.

-----Input-----
The first line has one integer $L$ indicating the number of square meters for Ellen’s lawn. The second line contains two integers $x_1$ and $y_1$, followed by a third line containing two integers $x_2$ and $y_2$. These indicate that the wall passes through the points $(x_1, y_1)$ and $(x_2,y_2)$.

-----Output-----
Output the chain length, in whole meters, which Tom needs to acquire.

-----Limits-----
 - $1 \leq L \leq 10000$
 - $-10000 \leq x_1, y_1, x_2, y_2 \leq 10000$
 - $(x_1,y_1) \not= (x_2,y_2)$

-----Examples-----
Sample Input:
4
-10 0
-10 10
Sample Output:
2
user: The Mischievous Troublemakers’ Association (MTA) has hatched a sinister plot for world domination! All through the night, the Troublemakers work tirelessly, arranging rocks in the shape of hideous, misshapen blobs on the lawns of the unsuspecting residents of Quackville. For the sake of efficiency, the MTA splits up into several factions, each of which covers a particular neighbourhood. The faction in charge of spreading doom to a given neighbourhood never visits the same lawn twice. While there is no consistency whatsoever as to the shapes of the evil rock blobs, the MTA is very specific about the number of rocks each blob should contain. In order to generate this number easily, the $k^{\mathrm{th}}$ faction ($k \geq 1$) has devised a sequence, $F_ k(n)$, whose $n^{\mathrm{th}}$ term ($n \geq 1$) is the number of rocks its members will use to make their blob on the $n^{\mathrm{th}}$ lawn they visit. This sequence can be defined recursively by\begin{align*} F_ k(1) & = 42\\ F_ k(2) & = 11k + 77\\ F_ k(n) & = 2F_ k(n - 1) - F_ k(n - 2) + 10k , \ \textrm{ for } n > 2 \end{align*}

So, for example, the members of the first faction ($k = 1$) place $42$ rocks on the first lawn they visit, $88$ rocks on the second lawn they visit, then $144$, $210$, $286$, and so on, on subsequent lawns.

While others stare confusedly at the amorphous piles of rocks, wondering what any of this has to do with world domination, Rex Tangleton leaps into action! Despite a complete lack of evidence, he is convinced that Quackvillians can only be saved if someone turns these ghastly blobs of doom into wonderful rectangles. According to Rex, a $p$ by $q$ rectangle of rocks is wonderful if $p$ and $q$ are prime. Note that he does not care whether or not $p$ and $q$ are equal. For each lawn that he visits, he will rearrange all of the rocks in the blob into a wonderful rectangle if this is possible. Otherwise, he will simply stare at the rocks and grunt, but not move any of them.

Originally, Rex had been planning to check as many lawns as possible. However, he has been informed that his grunting frightens people, and he will therefore not be paid for his work. Saving your community is no fun if you do not get remunerated, so Rex decides to pick random positive integers $k$ and $n$, and then check only the first $n$ lawns that the $k^{\mathrm{th}}$ faction has visited. Rex wants to know how much work lies ahead of him, so he has asked you to write a program that, given $k$ and $n$, will determine on how many lawns he will rearrange rocks.

-----Input-----
The input consists of one line with two space-separated positive integers, $k$ and $n$. It is guaranteed that $k \leq 100$ and $n \leq 1000000$.

-----Output-----
Output a single non-negative integer giving the number of lawns that Rex will visit on which he will rearrange rocks.

-----Examples-----
Sample Input:
1 5
Sample Output:
0
user: Yraglac has just finished building a robot! It’s fancy: two wheels, with an auto-balancing system, several cameras, wireless connectivity, and a sattelite dish. There’s just one little problem…

He powered it up, sent it off for a test, and now it’s not responding to any inputs. All he has is a photograph of the floor below it, acquired during an attempted rendition of Robot Ballet. Yraglac wasn’t really paying attention to where it was, though, as he was too excited.

Thankfully, one of Yraglac’s classmates has produced a huge stitched-together image of all of the floors in the University. Now all Yraglac has to do is figure out where this image of the floor is, and then he can just go and pick up the robot directly.

Unfortunately, Yraglac doesn’t have the patience to look through the entire huge image for anything that matches, so he wants to write a program that will find an initial set of locations that seem like candidates.

Thankfully, due to …cough …the fact that this is a competitive programming problem …cough …the images are of exactly the same scale, so Yraglac should be able to figure out where the robot is simply by checking how closely the robot’s image matches the floor image at each possible place to overlay the robot’s image on the floor image.

Yraglac will consider a location as a candidate if it is tied for having the most number of pixels the same between the two images.

-----Input-----
The input consists of two images, the first of which is the last image seen from the robot, and the second is the entire floor of the university.

Each image begins with a line consisting of two integers, $W$ and $H$, the width and height, respectively. You may assume that $W, H \leq 1000$, and that the floor image is at least as large as the robot’s image.

Following on the next $H$ lines are $W$ space-separated integers, either 0 or 1, denoting the colour at the given pixel.

-----Output-----
The output should consist of at least one line, one each for a candidate location, each containing a coordinate $(x,y)$ as two numbers separated by a single space, with $x$ first and $y$ second. The coordinates should be sorted by x-coordinate, and then by y-coordinate.

-----Examples-----
Sample Input:
2 2
1 0
0 1
3 3
1 0 0
0 1 0
0 0 1
Sample Output:
0 0
1 1
user: A certain big IT company (which we will not name in order not to get sued) is preparing to launch a new version of its flagship product. Having just been employed as a developer on the project, you have been given a list of open bugs that should be fixed in the new version. 

Being bugs, you are not exactly certain how to fix them, even though you have some ideas. For each bug you are able to estimate your ability to quickly fix the bug. Of course, these estimates may be wrong, so if you try to fix a bug and fail, you will revise the estimate of your ability to fix this bug.

To be specific, we use the following probabilistic model for the bug fixing process: for each bug, there is an associated fix probability $p$. Every hour, you choose one bug to work on, and work on this bug for the entire hour (if you manage to fix the bug in less then an hour, you celebrate by having coffee and taunting your coworkers for the remaining part of the hour). The probability that you succeed in fixing the bug during this hour is $p$. If you fail to resolve the bug, the fix probability for this bug is reduced to $p \cdot f$, where $f \le 1$ is a factor indicating how much faith you lose in your ability after a failure. The fix probabilities for the other bugs remain unchanged. The next hour, you again choose an open bug to work on, and so on. This is repeated until the new version is released, and you are allowed to go home and sleep.

In addition, each bug has a severity $s$ indicating how severe the bug is (or alternatively, the value of fixing the bug). Clearly, it is possible that you will not manage to fix all the bugs before the product is released. In order to make as good an impression on your boss as possible, you would like to maximize the total severity of those bugs which you do manage to resolve, by carefully choosing which bugs to work on. What will be the expected value of the total severity of fixed bugs, provided that you, every hour, choose which bug to work on in such a way that this quantity is maximized?

-----Input-----
The first line of input contains three numbers $B$, $T$ and $f$, where $0 \le B \le 10$ is an integer giving the number of open bugs, $0 \le T \le 100$ is an integer giving the number of hours left until the new version is released, and $0 \le f \le 1$ is a real number as described above.

Each of the following $B$ lines describe an open bug. Each such description contains two numbers $p$ and $s$, where $0 \le p \le 1$ is a real number giving the initial fix probability of the bug and $0 \le s \le 10000$ is an integer giving the severity of the bug.

All real numbers are given with exactly $6$ digits after the decimal point.

-----Output-----
Output a line containing the expected total severity of bugs fixed, provided you work in a way which maximizes this quantity. Any answer with either absolute or relative error smaller than $10^{-6}$ is acceptable.

-----Examples-----
Sample Input:
1 2 0.950000
0.700000 50
Sample Output:
44.975