meetforever/GRPO_CL huggingface dataset:
user: You are a seasoned Fight Club member who recently stumbled upon the classified plans of Project Mayhem, a secret organization that the other members have created to bring down consumerism and corporate America. To completely destabilize modern civilization, Tyler Durden and his Project Mayhem trainees are making plans to take down the buildings of major banks that hold everybody’s credit information. If they succeed, then everybody is supposed to restart life with a blank slate, free of all debts ranging from mortgages to student loans.


Tyler plans to do so by setting off pipe bombs under major financial buildings. His crew has mapped out the city’s financial district on a rectangular grid with $R$ rows and $C$ columns. Each grid cell either has a building armed with bombs to detonate (denoted by “x") or does not (denoted by “."). As a friend of Tyler’s, you know all too well that his plan is off the chains. Violence is never the solution. You want to do your best to thwart the plan by visiting the buildings one by one and disarming as many bombs as possible.


You can disarm the buildings in any order, with only one catch. After you disarm a building, Tyler’s crew will find out, chase after you, and occupy that building indefinitely. To avoid getting caught, you must immediately find another (armed) building in the same street or avenue to take cover before disarming other buildings. After that however, you can go to any other armed building on the map to disarm it (provided the same condition is met). In other words, at any time you disarm a building, there must be at least one other armed building in either the same row or the same column.


Take the following $3$ by $3$ map for example:

     x.. .x. x.x

You have a few ways to disarm up to $2$ buildings:
 - You can first disarm the top-left one, followed by either of the bottom two buildings, for a total of $2$ buildings.
 - You can first disarm the bottom-right one, followed by either of the remaining two leftmost buildings, for a total of $2$ buildings.
 - However, note that if you choose to first disarm the bottom-left building, then none of the remaining buildings can be disarmed (since you won’t be able to take cover afterwards). So in this case, you would only be able to disarm a single building.

In any of these cases, the center building can never be disarmed since there is no immediate neighboring building in its row or column to use for cover from Tyler’s henchmen.


Given a map of the city, you would like to know the maximum number of buildings that can be disarmed.

-----Input-----
The first line of input consists of two space-separated integers $R$ and $C$ ($1 \le R, C \le 2000$).

$R$ lines follow, each of which consists of $C$ characters (either “x" or “."), specifying the grid of the financial district.

-----Output-----
Print, on a single line, the maximum number of buildings that can be disarmed.


-----Examples-----
Sample Input:
3 3
x..
.x.
x.x
Sample Output:
2
user: Alice and Bob dream about future summer vacations where travel is unconstrained by pandemics, budgetary constraints, or other mundane problems. Since they love winter, they want to visit the Southern Hemisphere, in particular Chile. Chile is, geographically speaking, a rather one-dimensional country, and can be represented as an infinitely long $y$-axis. There are $N$ cities in Chile, numbered from $1$ to $N$, where the $i$th city is located at $y$-coordinate $y_ i$. Alice and Bob will start their journey in city number $1$, and travel to some other city from there.

Each city has an airport with a direct connection to other cities. However, flights cannot be arbitrarily short: For each city $i$ there is a minimal distance $d_ i$ such that there is a flight to city $j$ if and only if $|y_ i-y_ j|\geq d_ i$. The time for a flight is $|y_ i - y_ j|$ minutes, but it also takes time to travel to the airport, check in, pass security, etc. Thus, for each city there is an additional time $r_ i$ such that the total time to fly from $i$ to $j$ is $r_ i + |y_ i - y_ j|$ minutes.

Find the shortest possible time to get to each of the other cities, provided that Alice and Bob start in city $1$. Flights are the only means of transportation, but Alice and Bob can take as many flights as they like.

-----Input-----
The first line contains an integer $N$, the number of cities, where $2 \leq N \leq 2 \cdot 10^5$. The following $N$ lines each consist of three integers $y_ i$, $d_ i$, and $r_ i$, where $0 \leq y_ i, d_ i, r_ i \leq 10^9$. All $y$-coordinates are distinct.

-----Output-----
Print $N-1$ lines, each containing an integer. On the $i$th line, print the shortest possible total time for getting from city 1 to city $i+1$, if possible. If there is no way to get to city $i+1$, print $-1$ instead.

-----Examples-----
Sample Input:
5
1 3 2
2 5 2
3 0 0
4 2 4
5 3 0
Sample Output:
9
-1
5
6
user: The classic video game “Prince of Python” comprises $n$ levels, numbered from $1$ to $n$. You are going to speedrun this game by finishing all of the levels as fast as possible, and you can beat them in any order that you want.

You enter each level equipped with one of $n+1$ magical items. In the beginning you only have item $0$ in your inventory. Once you beat a level, you get to keep the item numbered the same as that level. For example, on finishing level $5$, you obtain a mighty Gauntlet of 5 Fingers you may equip thereafter instead of the less-acclaimed Sword of 0 Damage you always start out with.

Beating a level can take different amounts of time depending on which item you take into the level with you. Higher-numbered items are more powerful, so if playing by the rules it is always at least as fast to finish the level with a higher-numbered item as with a lower-numbered item.

However, each level also has a shortcut left in by the developers. The shortcut for a level can be accessed by applying a specific item in an unconventional way. By doing so you can finish the level as fast as, or even faster than, if you had used any of the other items.

How long will it take you to beat all of the levels of the game?

-----Input-----
The input consists of:
 - One line containing an integer $n$ ($1 \le n \le 2500$), the number of levels.
 - $n$ lines, describing the levels.

The $i$th such line starts with two integers $x_ i$ and $s_ i$ ($0 \le x_ i \le n$, $1 \le s_ i \le 10^9$), the shortcut item for level $i$ and the completion time for level $i$ when using the shortcut.

The remainder of the line has $n+1$ integers $a_{i,0}, \ldots , a_{i,n}$ ($10^9 \ge a_{i,0} \ge a_{i,1} \ge \ldots \ge a_{i,n} \ge s_ i$), where $a_{i,j}$ is the completion time for level $i$ when playing by the rules using item $j$.

-----Output-----
Output the minimum time it takes to beat, in any order, all of the levels in the game.

-----Examples-----
Sample Input:
3
1 1 40 30 20 10
3 1 95 95 95 10
2 1 95 50 30 20
Sample Output:
91
user: BAAS Inc. (Build Airplanes As-a-Service) has grand plans to build lots of airplanes. Naturally, the airplane construction process consists of multiple steps, each of which may take some time. Some steps can take more time to complete than others, e.g. attaching a plane wing may take longer than spray-painting the BAAS logo on the tail. Furthermore, some steps must to be completed before others can commence. For instance the hull needs to be assembled before the wings can be attached to it. If there is no dependency between two steps, then the steps can be completed in parallel.

BAAS is trying very hard to become the market leader in airplane construction. Since customers value a speedy delivery, BAAS has decided to invest in their R&D department with the goal to decrease their delivery time. The chief scientist of BAAS is very optimistic and thinks that his department can reduce the time it takes to complete one step to $0$. However, since the money is always tight in the research sector at BAAS, he could only promise to eliminate exactly one of the construction steps. As the CEO of BAAS, your role implies figuring out which step you would like to eliminate to achieve the shortest possible construction time. It goes without saying that as the CEO you have complete access to the BAAS’s secret construction process blueprints.

-----Input-----
The first line of the input contains an integer $N$ ($2 \le N \le 400$), the number of steps in the build process.

The next line contains the $N$ integers $a_1, a_2, \dots , a_ N$ ($1 \le a_ i \le 10^5$), the number of seconds each of the $N$ steps takes in the build process.

This is followed by $N$ lines. The $i$’th of these lines contains an integer $C_ i$ followed by $C_ i$ integers $A_1, \dots , A_{C_ i}$ ($1 \le A_ j < i$). The integers $A_ j$ denote what steps the $i$’th step has a dependency on.

It is guaranteed that there is no step that indirectly depends on itself. Furthermore, each step has a direct or indirect dependency on step $1$ (receiving an order), and step $N$ (delivering the airplane) has a direct or indirect dependency on every other step.

-----Output-----
Output a single integer – the shortest possible time it takes to construct an airplane assuming a single step can be reduced to take no time at all.

-----Examples-----
Sample Input:
2
15 20
0
1 1
Sample Output:
15
user: This is a very exciting week for John. The reason is that, as a middle school teacher, he has been asked to dedicate the entire week to teaching his class of $n$ students about the cause and effect of global warming. As John is very passionate about his planet, he’s going to spend extra time and effort to make this week memorable and rewarding for the students. Towards that, one of the things he wants to ask them to do is to prepare, as homework, presentations about global warming. To make this a little easier for them, as well as more fun, he has asked them to do this in groups of two.

Of course arranging the students into groups comes with the usual headache, namely that only friends are willing to work together. Luckily the students in his class are a friendly bunch. In particular, if $p$, $q$ and $r$ are three distinct students, and $p$ and $q$ are friends, and $q$ and $r$ are friends, then $p$ and $r$ are also friends. But John now realizes the irony in asking his students to work at home in groups, as students may have to travel to meet their group partner, which may emit greenhouse gases such as carbon dioxide, depending on their mode of transportation. In the spirit of this week’s topic, John asked all the students in his class to calculate, for each of their friends, how much carbon dioxide would be emitted if they were to meet up with the respective friend.

Using this information, can you help John figure out what is the minimum total amount of carbon dioxide that will be emitted if he arranges the groups optimally, or determine that it’s not possible to arrange all the students into groups of two friends?

-----Input-----
The first line contains two integers $n$ and $m$ ($1 \leq n \leq 200$, $0 \leq m \leq 250$), the number of students in John’s class, and the total number of pairs of friends in the class. As John is bad with names, he has given each of his students a distinct integer identifier between $1$ and $n$.

Each of the next $m$ lines contains three integers $p$, $q$ and $c$ ($1 \leq p, q \leq n$, $0 \leq c \leq 10^6$), the identifiers of two distinct students that are friends, and how many grams of carbon dioxide would be emitted if they were in a group together, and thus had to meet. Each pair of friends is listed exactly once in the input.

-----Output-----
Output the minimum total amount of carbon dioxide, in grams, that would be emitted if John arranges all students optimally into groups of two friends, or “impossible” if there is no way to arrange the students into groups in that way.

-----Examples-----
Sample Input:
5 4
3 1 375
2 5 283
1 4 716
3 4 98
Sample Output:
impossible
user: The game of Memory is played with $N$ pairs of cards where each pair has the same picture, i.e. there are $N$ different pictures, and each of them appear on exactly two cards.

The cards are shuffled and placed face down on a large table. On each turn you flip two cards of your choice face up. If they match you remove them from the game, and if they don’t match, you turn them face down. The goal of the game is to remove all cards from the game.

Your strategy is simple:
 - If you know of two face-down cards which have the same picture, choose both of them.
 - Otherwise,
 - Turn a random card that you have not looked at before face up.
 - If it matches a card you have seen, turn the matching card face up (you have excellent memory).
 - If the first card did not match a card you had seen before, turn another random unknown card face up.

Given this strategy, what is the expected number of turns you have to play in order to finish the game?

-----Input-----
Input is a single integer $N$ indicating the number of pairs of cards.

-----Output-----
Output the expected number of turns needed to finish the game. This number must have an absolute or relative error of at most $10^{-6}$.

-----Limits-----
 - $1 \leq N \leq 1000$

-----Examples-----
Sample Input:
1
Sample Output:
1.000000000
user: Everyone wants to live as long a life as possible. As time progresses, technology progresses. Various anti-aging pills get introduced to the market at various times which allow a person to age more slowly than normal. In particular, an $x$-$y$ pill, when taken regularly, ages your body only $y$ seconds over the course of $x$ seconds. So, if you took a $100$-$87$ pill regularly, then over the next $100$ seconds, your body would only age $87$ seconds. You can only take one of these pills at a time (or none at all). The only downside to using one of these pills is that due to the change in regimen, if you switch to a pill, it automatically ages you $c$ seconds. The value of $c$ is the same for all pills.

Any time you switch to an $x$-$y$ pill, you can take it for any number of seconds, but you can only switch to a pill at or after the time it first becomes available on the market. For the purposes of this problem assume that your life starts at time $t = 0$ seconds and that without the aid of any pills, you would live to be $n$ seconds old.

Given information about each different pill introduced into the market (the time it is introduced, in seconds, and its corresponding $x$ and $y$ values as previously described) and $c$, the number of seconds you automatically age when switching pills (or switching to a pill from no pill at all), determine the longest you can live, over all possible schedule of pills you could take.

-----Input-----
The first line of input consists of three positive integers, $n$ ($n \le 3\cdot 10^9$), representing the number of seconds you would live without taking any pills, $p$ ($p \le 10^5$), the number of pills that become available on the market, and $c$ ($c \le 10^5$), the time in seconds you age as soon as you switch to a different pill. $p$ lines follow with the $i^{th}$ line containing three space separated integers: $t_{i}$ $(1 \le t_{i} \le 10^{12})$, $x_{i}$ and $y_{i}$ $(1 \le y_{i} < x_{i} \le 10^{4}$), representing the time the $i^{th}$ pill gets introduced to the market, and the corresponding $x$ and $y$ values for it. In addition, for all $i$, $1 \le i \le n-1$, it is guaranteed that $t_{i+1} - t_{i} > c$.

-----Output-----
Output a single real number, representing the maximum number of seconds that you could live, if you take the appropriate pills. Your answer should be correct within a relative or absolute error of $10^{-6}$.

-----Examples-----
Sample Input:
100 3 10
15 99 98
40 3 2
90 10 9
Sample Output:
115.000000000
user: You are planning to travel in interstellar space in the hope of finding habitable planets. You have already identified $N$ stars that can recharge your spaceship via its solar panels. The only work left is to decide the orientation of the spaceship that maximizes the distance it can travel.

Space is modeled as a $2$D plane, with the Earth at the origin. The spaceship can be launched from the Earth in a straight line, in any direction. Star $i$ can provide enough energy to travel $T_ i$ distance if the spaceship is launched at an angle of $a_ i$ with the $x$-axis. If the angle is not perfectly aligned, then the spaceship gets less energy. Specifically, if the launch direction makes an angle of $a$ with the $x$-axis, then it gets enough energy to travel distance of\[ \max (0, T_ i - s_ i \cdot \mathrm{dist}(a_ i, a)) \]

from star $i$, where $\mathrm{dist}(a,b)$ is the minimum radians needed to go from angle $a$ to $b$. The distance that the spaceship can travel is simply the sum of the distances that each star contributes. Find the maximum distance $T$ that the starship can travel.

-----Input-----
The first line contains the value $N$, $1\le N\le 10^5$. Following this are $N$ lines each containing three real numbers $T_ i$, $s_ i$, and $a_ i$, with $0<T_ i\le 1000$, $0\le s_ i\le 100$, and $0\le a_ i < 2\pi $. All real numbers in the input have at most $6$ digits after the decimal point.

-----Output-----
On a single line output the maximum distance the spacecraft can travel. Your answer is considered correct if it has an absolute or relative error of at most $10^{-6}$.

-----Examples-----
Sample Input:
2
100 1 1
100 1 1.5
Sample Output:
199.500000
user: You are building advanced chips for machines. Making the chips is easy, but the power supply turns out to be an issue since the available batteries have varied power outputs.

Consider the problem of $n$ machines, each with two chips, where each chip is powered by $k$ batteries. Surprisingly, it does not matter how much power each chip gets, but a machine works best when its two chips have power outputs as close as possible. The power output of a chip is simply the smallest power output of its $k$ batteries.

You have a stockpile of $2nk$ batteries that you want to assign to the chips. It might not be possible to allocate the batteries so that in every machine both chips have equal power outputs, but you want to allocate them so that the differences are as small as possible. To be precise, you want to tell your customers that in all machines the difference of power outputs of the two chips is at most $d$, and you want to make $d$ as small as possible. To do this you must determine an optimal allocation of the batteries to the machines.

Consider Sample Input 1. There are $2$ machines, each requiring $3$ batteries per chip, and a supply of batteries with power outputs $1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12$. You can, for instance, assign the batteries with power outputs $1, 3, 5$ to one chip, those with power $2, 4, 12$ to the other chip of the same machine, those with power $6, 8, 9$ to the third chip, and those with power $7, 10, 11$ to the fourth. The power outputs of the chips are $1, 2, 6$, and $7$, respectively, and the difference between power outputs is $1$ in both machines. Note that there are many other ways to achieve this result.

-----Input-----
The input consists of a single test case. A test case consists of two lines. The first line contains two positive integers: the number of machines $n$ and the number of batteries per chip $k$ ($2nk \leq 10^6$). The second line contains $2nk$ integers $p_ i$ specifying the power outputs of the batteries ($1 \leq p_ i \leq 10^9$).

-----Output-----
Display the smallest number $d$ such that you can allocate the batteries so that the difference of power outputs of the two chips in each machine is at most $d$.

-----Examples-----
Sample Input:
2 3
1 2 3 4 5 6 7 8 9 10 11 12
Sample Output:
1
user: When Per was living in New York, he couldn’t use his debit card as often as in Sweden. He had to carry cash all the time and his wallet was getting heavy from all the coins he was carrying around. He was once in a candy store buying a few kilos of candy and he wanted to get rid of as many coins as possible but couldn’t figure out how to do it without getting any change back.

-----Task-----
You will be given a price $P$ that Per was going to pay. You will also be given the number of 1-, 5-, 10- and 25-cent coins that he had in his wallet. He didn’t have any notes. Find the maximum number of coins he could have used for paying the price $P$ without getting any change back.

-----Input-----
The first line of input contains an integer $P, 1\leq P\leq 100000000$, the price Per was going to pay. On the second line there are 4 space separated integers $N_1, N_5, N_{10}, N_{25}; 0\leq N_1, N_5, N_{10}, N_{25}\leq 100000000$, the number of 1-, 5-, 10- and 25-cent coins in Per’s wallet.

-----Output-----
If Per couldn’t pay exactly $P$ without getting any change back, output Impossible on a single line. Otherwise output the maximum number of coins Per could use to pay the price $P$.

-----Examples-----
Sample Input:
13
3 2 1 1
Sample Output:
5
user: Anthony has two multiple choice assignments to hand in today. The answer to each assignment is a string, where the $i^\textrm {th}$ letter represents the answer to the $i^\textrm {th}$ question.

But instead of handing in two strings, Anthony only hands in a single string. He claims he answered every question correctly but his little sister Cora merged his two strings into a single string. Moreover, Cora merged his string in such a way that the following conditions are satisfied:
 - For any two integers $0\leq i<j<|s_1|$ (where $|s_1|$ is the length of $s_1$), the index of $s_1[i]$ in $s$ is less than the index of $s_1[j]$ in $s$
 - For any two integers $0\leq i<j<|s_2|$, the index of $s_2[i]$ in $s$ is less than the index of $s_2[j]$ in $s$

Can you help Anthony’s teacher verify whether Anthony’s claim is possible?

-----Input-----
The first line contains a single string $s$. It is guaranteed that $2\leq |s|\leq 10000$.

The next two lines contain strings $s_1$ and $s_2$ respectively. It is guaranteed that $1\leq |s_1|, |s_2|\leq 5000$ and $|s_1|+|s_2|=|s|$.

All the strings consists of only lowercase English letters.

-----Output-----
If Anthony’s claim is possible, print “yes” (without quotes). Otherwise, print “no” (without quotes).

-----Examples-----
Sample Input:
aabcad
aba
acd
Sample Output:
yes