meetforever/GRPO_CL huggingface dataset:
user: Barry and Bruce are twin brothers. Bruce likes keeping his parenthetical sequences balanced. Barry would like to mess with Bruce by performing some operations on the sequence. Each operation is one of the following:
 - Change a single ‘(’ to a ‘)’ in the sequence.
 - Change a single ‘)’ to a ‘(’ in the sequence.

Bruce will attempt to rebalance the parenthetical sequence by performing the same operations. Bruce does not like tedium and will perform no more than $k$ operations to balance the sequence.

A balanced parenthetical sequence is defined as:
 - The empty string
 - $AB$ where $A$ and $B$ are both balanced parenthetical sequences
 - ($A$) where $A$ is a balanced parenthetical sequence

Barry would like to disrupt the sequence to the point where it is impossible for Bruce to rebalance the sequence in $k$ moves. Changing some position in the sequence requires effort and the amount of effort varies by position. Some positions are even delightful to switch and require negative effort. Each position can be changed at most once.

Barry hates effort and would like to compute the minimum sum of effort to ensure that Bruce cannot balance the sequence.

-----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 two integers $n$ and $k$, where $n$ ($1 \le n \le 10^5$) is the length of the sequence, and $k$ ($0 \le k \le n$) is the maximum number of moves for Bruce.

The next line contains a single string of length $n$ consisting of only the characters ‘(’ and ‘)’. This string is NOT required to be balanced.

The next $n$ lines will each contain a single integer $c$ ($-1000 \le c \le 1000$), which is the cost of changing each parenthesis in order.

-----Output-----
Output a single integer, which is the minimum sum of effort of moves required to make the string impossible to be balanced by Bruce. If Bruce can always rebalance the string regardless of Barry’s actions, print a single question mark (‘?’).

-----Examples-----
Sample Input:
4 1
((()
480
617
-570
928
Sample Output:
480
user: Jay runs a small carnival that has various rides and attractions. Unfortunately, times are tough. A recent roller coaster accident, flooding in the restrooms, and an unfortunate clown incident have given Jay’s carnival a bad reputation with the public. With fewer paying customers and reduced revenue, he will need to cut some costs to stay in business.

One of the biggest carnival attractions is a large, confusing maze. It consists of a variety of circular rooms connected by narrow, twisting corridors. Visitors love getting lost in it and trying to map it out. It has come to Jay’s attention that some of the rooms might be effectively identical to each other. If that’s the case, he will be able to reduce its size without anyone noticing.

Two rooms $A$ and $B$ are effectively identical if, when you are dropped into either room $A$ or $B$ (and you know the map of the maze), you cannot tell whether you began in $A$ or $B$ just by exploring the maze. The corridor exits are evenly spaced around each room, and you cannot mark or leave anything in a room (in particular, you cannot tell whether you have previously visited it). The only identifying feature that rooms have is their number of exits. Corridors are also twisty enough to be indistinguishable from each other, but when you enter a room you know which corridor you came from, so you can navigate a little by using the order they appear around the room.

Jay has appealed to the Association for Carnival Mazery for help. That’s you! Write a program to determine all the sets of effectively identical rooms in the maze.

-----Input-----
The input consists of a single test case. The first line contains an integer $n$, the number of rooms in the maze ($1 \leq n \leq 100$). Rooms are numbered from 1 to $n$. Following this are $n$ lines, describing each room in order. Each line consists of an integer $k$, indicating that this room has $k$ corridors ($0 \leq k < 100$), and then $k$ distinct integers listing the rooms each corridor connects to (in clockwise order, from an arbitrary starting point). Rooms do not connect to themselves.

-----Output-----
Display one line for each maximal set of effectively identical rooms (ignoring sets of size 1) containing the room numbers in the set in increasing order. Order the sets by their smallest room numbers. If there are no such sets, display none instead.

-----Examples-----
Sample Input:
13
2 2 4
3 1 3 5
2 2 4
3 1 3 6
2 2 6
2 4 5
2 8 9
2 7 9
2 7 8
2 11 13
2 10 12
2 11 13
2 10 12
Sample Output:
2 4
5 6
7 8 9 10 11 12 13
user: Spike is competing in the Gauntlet of Fire for the title of Dragon Lord. To succeed, he needs to traverse the labyrinthine volcanic tunnels, avoiding the treacherous elements and making it from chamber to chamber as quickly as possible.

The cave has $N$ chambers and $M$ tunnels connecting these chambers; no tunnel connects a chamber to itself, and there is at most one tunnel between any two chambers. This cave has a special property: it is known that it is possible to travel between any two different chambers by taking one or more tunnels, and that there are at most two paths between any two different chambers that do not take any tunnel more than once.

For simplicity, we label the chambers $1, 2, \dots , N$; then, the $i^\text {th}$ tunnel connects chambers $A_ i$ and $B_ i$, and has a length of $L_ i$ meters.

Spike can travel at the fast, fast speed of $1$ meter per second. He wants to know the danger level of each chamber. The danger level of a chamber is simply the sum of the times, in seconds, required to travel from this chamber to every other chamber, assuming that one always takes the shortest path possible.

Help Spike determine the danger level of each chamber!

Since these numbers can be quite large, you should output only the remainders after dividing each number by $10^9+7$.

-----Input-----
The first line of input contains two integers, $N$ ($2 \leq N \leq 200000$) and $M$ ($N-1 \leq M \leq 400000$) the number of chambers and tunnels in the cave.

The next $M$ lines contain the descriptions of the tunnels. In particular, the $i^\text {th}$ of these lines contains three integers $A_ i$, $B_ i$ ($1\leq A_ i, B_ i \leq N$; $A_ i \neq B_ i$) and $L_ i$ ($1 \leq L_ i \leq 10^9$), denoting that the $i^\text {th}$ tunnel connects chambers $A_ i$ and $B_ i$, and has a length of $L_ i$ meters.

It is guaranteed that there is at most one tunnel between any two chambers, that it is possible to travel between any two different chambers by taking one or more tunnels, and that there are at most two paths between any two chambers that do not take any tunnel more than once.

-----Output-----
Output $N$ integers on a single line, separated by spaces. The $i^\text {th}$ of these integers should contain the danger level of chamber $i$.

Since these numbers can be quite large, you should output only the remainders after dividing each number by $10^9+7$.

-----Examples-----
Sample Input:
5 5
1 2 3
1 4 8
2 3 12
3 5 4
4 5 2
Sample Output:
35 39 36 27 29
user: Being a Pokenom trainer is very stressful. You always need to take good care of all your Pokenoms and always be prepared to battle other Pokenom trainers.

To cope with the stressful life, Bash learns to paint in his free time. Today Bash is working on his masterpiece: ‘The two staircases’.

The painting is divided into a grid with $M$ rows and $N$ columns. Rows and columns are numbered starting from $1$, from bottom to top and from left to right, respectively. Let $(i, j)$ denote the cell at the $i$-th row and $j$-th column.

Bash colors each cell red or blue, such that the bottom part forms a blue ‘staircase’ and the top part forms a red ‘inverted staircase’. More formally, Bash’s picture has the following properties:
 - For every column $i$ $(1 \leq i \leq N)$, there exists two integers $b_{i}, r_{i}$ satisfying:
 - $0 \leq b_{i}, 0 \leq r_{i}, b_{i} + r_{i} \leq M$.
 - $b_{i}$ bottommost cells (i.e, cells $(1, i), (2, i), \ldots , (b_{i}, i)$) are blue.
 - $r_{i}$ topmost cells (i.e, cells $(M, i), (M - 1, i), \ldots , (M - r_{i} + 1, i)$) are red.
 - All other cells are not painted.
 - $M \geq b_{1} \geq b_{2} \geq \ldots \geq b_{N} \geq 0$.
 - $0 \leq r_{1} \leq r_{2} \leq \ldots \leq r_{N} \leq M$.

Hence, Bash’s picture can be uniquely determined by two sequences $b = (b_{1}, b_{2}, \ldots , b_{N})$ and $r = (r_{1}, r_{2}, \ldots r_{N})$. This is an example of a valid picture with $M=5$, $N=4$, $b = (4, 2, 2, 0)$ and $r = (1, 1, 2, 3)$:

Below are three examples of invalid pictures:

After few hours of hard work, Bash has finished his painting, and shows his best friend Cee. The picture satisfies all the above properties, with parameters $b = (c_{1}, c_{2}, \ldots , c_{N})$ and $r = (M - c_{1}, M - c_{2}, \ldots M - c_{N})$. No cells are left unpainted in this picture.

Cee wants to know step-by-step of how Bash created such beautiful painting. Bash can not remember the order which he painted the cells, but Bash remembers that he always followed these rules:
 - Bash starts with an empty picture.
 - First, Bash paints the bottom-left cell $(1, 1)$ blue and the top-right cell $(M, N)$ red.
 - In each step, Bash chooses some unpainted cell, paints it either red or blue, such that the picture after this step satisfies all the above properties.
 - The process stops when the picture is completed.

Cee tries to follow Bash’s rules to replicate the painting. But first Cee wants to know how many ways Cee can create the painting. Two ways are considered different if there exists a step where the painted cells are different.

Represent the result as $100003^X \times Y$, where $Y$ is not divisible by $100003$, and output $X \; Y_ m$ where $Y_ m$ is the result of $Y \bmod 100003$.

-----Input-----
 - Line $1$: Two integers $N$ and $M$ $(1 \le M, N \le 1000)$.
 - Line $2$: $N$ integers $c_1, c_2, \cdots , c_ N$ $(M \ge c_1 \ge c_2 \ge \cdots \ge c_ n \ge 0)$ describes the blue parameters in Bash’s final picture (see description above).

-----Output-----
Two integers $X$ and $Y_ m$ as described above.

-----Sample clarification-----
Bash’s pictures in $2$ sample cases:

-----Examples-----
Sample Input:
3 3
3 2 1
Sample Output:
0 672
user: The Autonomous Car Manufacturer (ACM) needs to design algorithms to control their cars. One particular problem is lane switching—if the car needs to switch lanes, it wants to do so in the safest manner.

Initially the car is in the leftmost lane, and the car needs to switch to the rightmost lane. The car has a variety of sensors and can obtain the locations of all cars in a section of the highway. When the car needs to switch lanes, it takes a snapshot of the sensor readings and design a plan to switch lanes based on the snapshot. The sensors have limited range. All sensor readings will be distances from the start of the sensor range. For safety reason, the areas outside of the sensor range are assumed to be occupied by cars.

You may assume that all other cars are travelling at exactly the speed limit. However, the ACM would like to set itself apart by producing cars that may drive at any speed regardless of the speed limit, as long as it does not hit any other car. For safety reasons, a lane switch is always done while driving at the speed limit.

When a lane switch occurs, the destination must have unoccupied space for the car to move into (a perfect fit is allowed). We define the safety factor of the plan as the closest distance to any car while executing the plan. We are only concerned about cars in the same lane, and will ignore distances between cars in different lanes. Obviously, the ACM wants its cars to choose a plan that has the highest safety factor.

The first sample input is illustrated below.

-----Input-----
The first line of input contains three integers $N$ ($2 \leq N \leq 100$), $M$ ($M \geq 1$), $R$ ($1 \leq R \leq 1000000$) indicating the number of lanes, the number of cars on the road, and the sensor range. The next $M$ lines describe each car with three integers: the lane number (between 0 and $N-1$, inclusive), the length of the car (positive), and the distance from the start of the sensor range to the back of the car. The distance is non-negative and at most $R$. The first car given is the ACM car. Lane 0 is the leftmost lane, and lane $N-1$ is the rightmost lane.

There are at most 100 cars in each lane. No two cars will overlap although they may touch bumper-to-bumper.

-----Output-----
If the ACM car can switch from lane 0 to lane $N-1$, print a single number indicating the maximum achievable safety factor. Otherwise, print Impossible. Your answer will be considered correct if its absolute error does not exceed $10^{-5}$.

-----Examples-----
Sample Input:
4 5 100
0 10 10
1 10 5
1 20 35
2 2 18
2 40 50
Sample Output:
2.500000
user: You are an employee at Micah’s Airlift Payload Services (MAPS), which is a local crane company. On this particular day, you receive a call from a guy named Jerry, asking for help in moving a large number of wooden crates.

Jerry has $N$ stacks of crates, arranged side-by-side, in a perfectly straight row. However, not all of the stacks are of the same height, and Jerry is concerned that some of the taller stacks may tip over. To help ease his worries, Jerry has asked you to rearrange the crates so that each stack is composed of the same number of crates. In order to accomplish this goal, you are permitted to create new stacks of crates, however, he has asked you not to completely remove any of the existing stacks. A new stack can only be added immediately to the right of the current rightmost stack.

The crane is initially positioned above the leftmost stack of crates. As the crane operator, you are able to perform $3$ different types of actions. You can either pick up one crate from the top of the current stack, drop one crate onto the top of the current stack, or move the crane to an adjacent stack. The crane may not carry more than one crate at a time.

What is the minimum number of actions required to rearrange the crates into equal-sized stacks? The final position of the crane does not matter, however, the crane must not be left carrying a crate.

-----Input-----
The first line of input contains an integer $N$, where $2\leq N\leq 50000$, indicating the number of stacks. The next line contains $N$ space-separated integers, indicating the number of crates in each of the stacks, from left to right. Each stack contains between $1$ and $50000$ crates, inclusive.

-----Output-----
Output the minimum number of actions required to complete the job.

-----Examples-----
Sample Input:
3
5 3 4
Sample Output:
3
user: You’re playing hopscotch! You start at the origin and your goal is to hop to the lattice point $(N, N)$. A hop consists of going from lattice point $(x_1, y_1)$ to $(x_2, y_2)$, where $x_1 < x_2$ and $y_1 < y_2$.

You dislike making small hops though. You’ve decided that for every hop you make between two lattice points, the x-coordinate must increase by at least $X$ and the y-coordinate must increase by at least $Y$.

Compute the number of distinct paths you can take between $(0, 0)$ and $(N, N)$ that respect the above constraints. Two paths are distinct if there is some lattice point that you visit in one path which you don’t visit in the other.

Hint: The output involves arithmetic mod $10^9+7$. Note that with $p$ a prime like $10^9+7$, and $x$ an integer not equal to $0$ mod $p$, then $x(x^{p-2})$ mod $p$ equals $1$ mod $p$.

-----Input-----
The input consists of a line of three integers, $N$ $X$ $Y$. You may assume $1 \le X, Y \le N \le 10^{6}$.

-----Output-----
The number of distinct paths you can take between the two lattice points can be very large. Hence output this number modulo 1 000 000 007 ($10^9 + 7$).

-----Examples-----
Sample Input:
2 1 1
Sample Output:
2
user: As you might have heard, Gunnar is an old and forgetful researcher. Most of his research is in security and he cares a bit too much about his own security, so for each website he has a different password. It would be very hard for him to remember all passwords, so for every website he only remembers the method he used to create the password.

For one of the very important websites he started with a file containing a long list of non-negative integers. Since he very much likes the operation $\oplus $ (xor), his password is a xor of some integers in the list. Note that the operation xor is defined on boolean values as $0\oplus 0=1\oplus 1=0$ and $0\oplus 1=1\oplus 0=1$. We can then extend this definition to integers, namely we first write the two integers in binary and then do xor for each two corresponding bits in the numbers. For example the xor of $12=(1100)_2$ and $5=(101)_2$ is $9=(1001)_2$. Instead of addition, we can use the operation xor when summing numbers and we call this modified sum xor-sum.

-----Task-----
Gunnar’s file contains a list of numbers and he selected a subset of the numbers such that its xor-sum is as large as possible. The resulting number was his password. Unfortunately, he forgot the algorithm to find the subset with the largest xor-sum, so he is asking you for help with restoring his password. Of course, he will not tell you for which website this password is.

-----Input-----
The first line of input contains an integer $n$ $(1\le n\le 100000)$: the length of the list of numbers in Gunnar’s file. The second line contains $n$ space separated integers $a_1, \dots , a_ n$ $(1\leq a_ i \le 10^{18})$, the numbers in the file.

-----Output-----
Output one line with the answer – the maximal number Gunnar can get by selecting a subset of the list of numbers and calculating the xor-sum of the subset.

-----Examples-----
Sample Input:
3
1 3 5
Sample Output:
7
user: Each resident of a particular town is a member of zero or more clubs and also a member of exactly one political party. Each club is to appoint one of its members to represent it on the town council so that the number of council members belonging to any given party does not equal or exceed half the membership of the council. The same person may not represent two clubs; that is there must be a 1-1 relationship between clubs and council members. Your job is to select the council members subject to these constraints.

-----Input-----
In the first line of input there will be an integer $T \le 12$, giving the number of test cases.

Each of the $T$ testcases begins with a line with the number of residents $n$, $1 \leq n \leq 1000$. The next $n$ lines each contain a resident, a party, the number of clubs the resident belongs to (at most $110$) and the names of such clubs. Each resident appears in exactly one input line.

-----Output-----
For each test case, follow the following format: Each line should name a council member followed by the club represented by the member. If several answers are possible, any will do. If no council can be formed, print “Impossible.” in a line. There will be a blank line in between two test cases.

-----Examples-----
Sample Input:
2
4
fred dinosaur 2 jets jetsons
john rhinocerous 2 jets rockets
mary rhinocerous 2 jetsons rockets
ruth platypus 1 rockets
4
fred dinosaur 2 jets jetsons
john rhinocerous 2 jets rockets
mary rhinocerous 2 jetsons rockets
ruth platypus 1 rockets
Sample Output:
fred jetsons
john jets
ruth rockets

fred jetsons
john jets
ruth rockets
user: There is an old adage about opening a can of worms. A lesser known adage is one about shooting a can of exploding worms with a BB gun.

Imagine we place some cans of exploding worms on a long, straight fence. When a can is shot, all of the worms inside will explode. Different types of worms have different blast radii. Each can contains only one kind of worm.

When a can explodes, if another can is in the blast radius, then that can will also explode, possibly creating a chain reaction. Each can explodes only once. This process continues until all explosions stop.

For each can, suppose that it is the only can shot. How many cans in total will explode?

-----Input-----
There will be a single test case in the input. This test case will begin with a line with a single integer $n$ ($1 \le n \le 100000$) representing the number of cans on the fence. Each of the next $n$ lines will have two integers $x$ ($-10^9 \le x \le 10^9$) and $r$ ($1 \le r \le 10^9$), where $x$ is the location of the can on the fence and $r$ is the blast radius. No two cans will occupy the same location.

-----Output-----
Print $n$ integers on a single line separated by single spaces. The $i^{th}$ integer represents the number of cans that will explode if the $i^{th}$ can is the one that is shot.

-----Examples-----
Sample Input:
3
4 3
-10 9
-2 3
Sample Output:
1 2 1
user: You are implementing an operating system, and now need to write a program to list files in a directory: “ls”. You want the user to be able to list only files that match a given pattern that can include wildcards (*), for example *.c. A wildcard matches zero or more characters of any kind.

-----Input-----
The first line contains a string $P$, containing $1$–$100$ characters ’a’–’z’, ’*’ and ’.’. This is the pattern. The second line contains an integer $N$, $1 \leq N \leq 100$, which is the number of files in the directory. Then follow $N$ lines containing the names of the files in the directory. Each line is a string containing $1$–$100$ characters ’a’–’z’ and ’.’.

-----Output-----
The output shall consist of the filenames that match the pattern, $P$, each on its own line, in the same order that they were given as input.

-----Examples-----
Sample Input:
*.*
4
main.c
a.out
readme
yacc
Sample Output:
main.c
a.out
user: Anthony and Cora play a very exciting game. Initially, Anthony has $N$ points and Cora has $M$ points. The game then goes on for several rounds, during each round, either Anthony wins or Cora wins, the loser of the round loses $1$ point while nothing happens to the winner of the round. The game ends when either Anthony or Cora has no points left, and the one still left with points is the winner of the game.

Cora promises Anthony a sweet prize if he wins the game, but will cruelly humiliate Anthony if he loses. Anthony can very accurately evaluate himself and perfectly predict that his probability of winning the round $i$ is exactly $p_ i$. Now, in order to decide whether to play this game with Cora or not, he needs to know the probability of him winning the game.

Help Anthony find his probability of winning!

-----Input-----
The first line contain integers $1\leq N,M\leq 1000$. $N+M-1$ lines follow, with the $i$-th line containing $0\leq p_ i\leq 1$, $p_ i$ has at most $6$ decimal digits.

-----Output-----
Output a single line containing the probability of Anthony winning this game. Your answer will be considered correct if its absolute or relative error doesn’t exceed $10^{-6}$.

-----Examples-----
Sample Input:
1 1
0.500000
Sample Output:
0.500000