meetforever/GRPO_CL huggingface dataset:
user: You are given an matrix of $R$ rows and $C$ columns. All elements of the matrix have absolute value smaller than or equal to $10^4$. You may perform the following operations:

Operation

Notation

Example 

Rotate $i$-th row of the matrix $k$ elements right ($1 \leq i \leq R$, $1 \leq k < C$).

$\mathrm{rotR}\ i\ k$

$\mathrm{rotR}\ 3\ 1$

$ \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ 10 & 11 & 12 \end{pmatrix} \mapsto \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 9 & 7 & 8 \\ 10 & 11 & 12 \end{pmatrix} $

Rotate $j$-th column of the matrix $k$ elements down ($1 \leq j \leq C$, $1 \leq k < R$).

$\mathrm{rotS}\ j\ k$

$\mathrm{rotS}\ 3\ 1$

$ \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ 10 & 11 & 12 \end{pmatrix} \mapsto \begin{pmatrix} 1 & 2 & 12 \\ 4 & 5 & 3 \\ 7 & 8 & 6 \\ 10 & 11 & 9 \end{pmatrix} $

Multiply all elements in the $i$-th row by $-1$, if and only if none of them were multiplied before ($1 \leq i \leq R$).

$\mathrm{negR}\ i$

$\mathrm{negR}\ 2$

$ \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ 10 & 11 & 12 \end{pmatrix} \mapsto \begin{pmatrix} 1 & 2 & 3 \\ -4 & -5 & -6 \\ 7 & 8 & 9 \\ 10 & 11 & 12 \end{pmatrix} $

Multiply all elements in the $j$-th column by $-1$, if and only if none of them were multiplied before ($1 \leq j \leq C$).

$\mathrm{negS}\ j$

$\mathrm{negS}\ 1$

$ \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ 10 & 11 & 12 \end{pmatrix} \mapsto \begin{pmatrix} -1 & 2 & 3 \\ -4 & 5 & 6 \\ -7 & 8 & 9 \\ -10 & 11 & 12 \end{pmatrix} $Table 1: Allowed operations 

Using a limited number of these operations, you need to maximize the sum of all the elements of the matrix.

-----Input-----
The first line of input contains two integers $R$ and $C$ ($1 \leq R, C \leq 100$), the number of rows and columns. The next $R$ lines contain $C$ integers each. All integers are by their absolute value smaller than $10^4$.

-----Output-----
The first line of output should contain two integers, the maximal sum obtainable and the number of operations used. We shall call this number $T$, and it must hold that $T \leq 5RC$. The next $T$ lines should contain any sequence of operations leading to the sum. Each operation should follow the notation defined in Table 1. For details look at sample test cases.

-----Examples-----
Sample Input:
3 4
1 -2 5 200
-8 0 -4 -10
11 4 0 100
Sample Output:
345 2
rotS 2 1
negR 2
user: When simulating x-ray imaging different energy photons interact differently and therefore needs to be simulated separetly. Often one has an x-ray spectrum with thousands of bins, so that it is impractical to do a simulation for each bin. We want to do simulations only for a few energies, but want to choose these energies in an optimal way. Energies for the simulation should be chosen to minimize the sum over all photons of the square of the distance between the energy of that photon and its closest simulation energy.

-----Input-----
The input will start with two integers $1\leq n\leq 200$ and $1\leq m \leq n$ denoting the number of energies in the input spectrum and the number of simulation energies, respectively. This is followed by a line of $n$ integers $0\leq k_ i\leq 10^6$, where $k_ i$ ($1\leq i \leq n$) is the number of photons with energy $i$ in the x-ray spectrum. Simulation energies $E_ j$ ($1\leq j \leq m$) should be selected so as to minimize\[ \sum _{i=1}^n k_ i\min _ j\left((i-E_ j)^2\right). \]

-----Output-----
Output one number, the minimum value of the sum above, with a relative or absolute error less than $10^{-4}$.

-----Example-----
In sample input 1 we are have $n=3$, $m=2$, $k_1=3$, $k_2=1$, and $k_3=1$. Optimal simulation energies will be $E_1=1$ and $E_2=2.5$, since this will remove the contribution from bin 1 and minimize the contribution from bins 2 and 3. The sum will be $k_1(1-E_1)^2+k_2(2-E_2)^2+k_3(3-E_2)^2=3\cdot 0^2+1\cdot 0.5^2+1\cdot 0.5^2=0.5$.

-----Examples-----
Sample Input:
3 2
3 1 1
Sample Output:
0.5
user: Jon Dayne is the ruler of a huge country called Seven Kingdoms. He has two sisters, Arya and Sansa, and wants to give some cities of Seven Kingdoms to them. He will rule the remaining cities or if no city remains, goes to the Wall, a colossal fortification along the northern border of the Seven Kingdoms, to be the Lord commander. Arya is the Lady of Winterfell and Sansa is the Lady of King’s Landing. The cities in Seven Kingdoms (including Winterfell and King’s Landing) are connected to each other with a network of roads (although some cities may be disconnected from the other cities, because they are either located on an island or they are currently at war with these other cities). There is no direct road between Winterfell and King’s Landing and they do not share a common neighbour city.

Jon wants to assign a collection of cities to each one of his sisters such that each city in a collection is connected with a direct road to all other cities in that collection and the remaining cities, not in these two collections, are also connected with a direct road to each other. The collection assigned to Arya must include Winterfell and the collection assigned to Sansa must include King’s Landing. Jon needs your help to determine whether this is possible and if this is possible, you should tell him the cities in each collection.

-----Input-----
The input consists of a single test case. The first line contains two integers $n$ and $m$, where $n$ ($2 \le n \le 2000$) is the number of cities, and $m$ ($0 \leq m \leq m(m-1)/2$) is the number of roads. Each of the next $m$ lines contains two integers $x_ i$ and $y_ i$ ($1 \le x_ i, y_ i \le n$) describing one road, where $x_ i$ and $y_ i$ are the distinct cities the road connects. Two cities are connected by at most one road. Winterfell is city $1$ and King’s Landing is city $2$ in the road network.

-----Output-----
If it is not possible to partition the cities in the way explained, display the word impossible. Otherwise, display two lines: the first containing the cities in the collection assigned to Arya and the second containing the collection of cities assigned to Sansa. If there are many such collections, any one of them is acceptable.

-----Examples-----
Sample Input:
9 11
1 4
5 4
1 5
6 2
6 7
7 2
3 8
3 9
8 9
6 8
5 9
Sample Output:
1 4 5
2 6 7
user: You and your friends have gathered at your house to prepare for the Big Game, which you all plan to attend in the afternoon at the football stadium across town. The problem: you only have $k$ cars between you, with each car seating five people (including the driver), so you might have to take multiple trips to get all $n$ people to the stadium. In addition, some of your friends know the city better than others, and so take different amounts of time to drive to the stadium from your house. You’d like to procrastinate as long as possible before hitting the road: can you concoct a transportation plan that gets all people to the stadium in the shortest amount of time possible?

More specifically, each person $i$ currently at your house can drive to the stadium in $t_ i$ minutes. All $k$ cars are currently parked at your house. Any person can drive any car (so the cars are interchangeable). After a car arrives at the stadium, any person currently at the stadium can immediately start driving back to your house (and it takes person $i$ the same amount of time $t_ i$ to drive back as to drive to the stadium), or alternatively, cars can be temporarily or permanently parked at the stadium. Drivers driving to the stadium can take up to four passengers with them, but drivers driving back can NOT take any passenger. You care only about getting all $n$ people from your house to the stadium—you do NOT need to park all $k$ cars at the stadium, if doing so would require more time than an alternative plan that leaves some cars at your house.

-----Input-----
The first line of input contains two space-separated integers $n$ and $k$ $(1 \leq n,k \leq 20000)$, the number of people at your house and the number of available cars. Then follow $n$ lines containing a single integer each; the $i$th such integer is the number of seconds $t_ i$$(1 \leq t_ i \leq 1000000)$ that it takes person $i$ to drive from your house to the stadium, or vice-versa.

-----Output-----
Print the minimum number of seconds it takes to move all $n$ people from your house to the stadium, if all people coordinate and drive optimally.

-----Examples-----
Sample Input:
11 2
12000
9000
4500
10000
12000
11000
12000
18000
10000
9000
12000
Sample Output:
13500
user: Doug James is an up-and-coming DJ from Graphland who’s had a tough time making it big. This all changed with the release of his latest EP Wiggly Waves, which is the first album in history to go both Platinum and Uranium. With his newfound popularity, Doug (a.k.a. DJ Polygon) needs help with a problem most artists would be lucky to face: deciding which of his many gig offers to take.

There are $K$ venues in Graphland, connected by $R$ roads. The venues are numbered $1$ to $K$. Doug’s house is venue $1$, and he is ready to leave or perform at time $t = 0$.

Doug has $G$ gig offers. The $i$-th gig is at venue $V_ i$, runs during the time interval $[S_ i,E_ i)$ (inclusive start, exclusive end), and pays out $M_ i$ cryptocents. Doug can’t take multiple gigs at the same time, and he can’t take gigs while he’s traveling between venues.

Doug is overwhelmed by his newfound fame and many gig requests, and wants your help making as much money as possible.

-----Input-----
The first line of the input contains three integers $G$, $K$, and $R$: the number of gigs Doug has been offered, the number of venues in Graphland, and the number of roads connecting these venues.

These integers satisfy $1 \leq G \leq 200000$, $1 \leq K \leq 100$, and $0 \leq R \leq \min \left(4000, K(K-1)/2\right)$.

Then follow $R$ lines, each of which has three integers $A_ i$, $B_ i$, and $T_ i$, specifying the $i$-th (bidirectional) road. The $i$-th road connects venues $A_ i$ and $B_ i$ and it takes time $T_ i$ to travel the road in either direction. The values satisfy $1 \leq A_ i, B_ i \leq K$, $A_ i \neq B_ i$, and $1 \leq T_ i \leq 1000000$. Every road has two distinct endpoints, and there is at most one road between any pair of venues.

Then follow $G$ lines, each of which has four integers $V_ i$, $S_ i$, $E_ i$, and $M_ i$. This means the $i$-th gig runs from time $S_ i$ (inclusive) to $E_ i$ (exclusive) at venue $V_ i$, and pays $M_ i$ cryptocents. These values satisfy the bounds $0 \leq S_ i < E_ i \leq 1000000000$ and $1 \leq M_ i \leq 1000000$.

-----Output-----
Output a single integer: the maximum number of cryptocents that DJ Polygon can make by taking on the right gigs.

-----Sample Explanation-----
In the first sample, There are two gigs at venue $1$ and one gig at venue $2$. Doug can either play both gigs at venue $1$ for $11$ cryptocents, or spend the first $10$ units of time traveling to venue $2$ and play that gig for $33$ cryptocents. He chooses the latter.

In the second sample, Doug makes the most by staying at venue $1$ and playing both of those gigs to earn $70$ cryptocents.

-----Examples-----
Sample Input:
3 2 1
1 2 10
1 4 6 6
1 6 10 5
2 10 30 33
Sample Output:
33
user: Having just moved to Transylvania, one of the first orders of business is to get a transit card for the public transportation. The transit card can be charged with intervals of dates. For each such interval, the first few days come at a pretty high price per day, the next few days come at a slightly lower price per day, and so on. 

Since you miss your friends back home, you will frequently be going home for a few days. On the one hand, it seems wasteful to pay for keeping the transit card charged during the days you are away. But on the other hand, if you do not keep it charged then you have to start a new date interval when you get back and will be forced to pay the higher price for a few days again.

Given the pricing scheme and the schedule of which days you are going to be away, you decide to write a program to help you determine the cheapest option.

-----Input-----
The first line of input contains an integer $1 \le l \le 100$, the number of price levels for the transit card. Then follows a line containing $l$ integers $p_1, p_2, \ldots , p_{l}$, where $1 \le p_ i \le 1000$ is the price per day at the $i^\textrm {th}$ price level. The next line contains $l-1$ integers $d_1, d_2, \ldots , d_{l-1}$, where $1 \le d_ i \le 10^6$ indicates how many days the $i^\textrm {th}$ price level is active. Thus for example the third price level $p_3$ becomes active after $d_1 + d_2$ days. The last price level is active indefinitely. You may assume that the prices are monotonically decreasing, i.e., $p_1 > p_2 > \ldots > p_ l$.

The fourth line of input contains two integers $t$ and $n$, where $1 \le t \le 10^6$ is the total number of days for which you want to buy a transit pass, and $0 \le n \le 5000$ is the number of trips home you make during this period. After this follow $n$ lines, each containing two integers $a$ and $b$ ($1 \le a \le b \le t$), indicating that you will be away on all days from day $a$ to day $b$ (inclusive). The days are numbered from $1$ to $t$. You may assume that the trips home are listed in chronological order and that they do not overlap.

-----Output-----
Output the smallest total cost of your transit card for the $t$ days.

-----Examples-----
Sample Input:
3
20 15 10
7 7
30 0
Sample Output:
405
user: The Miniature Toy Association (or MTA for short) is releasing the latest edition of its brand of collectible figurines. MTA used to sell them individually, but now has decided to adopt a new business strategy that involves randomly choosing $4$ figurines from the entire collection to be packaged and sold together as a ‘$4$-pack’. Each figurine in a $4$-pack is chosen independently and randomly with uniform distribution. As such, it is possible that a $4$-pack may contain multiple copies of a particular figurine.

Even though some figurines may be more valuable than others, every randomly assembled $4$-pack is sold at the same price. Customers buying a $4$-pack of figurines do not know ahead of time which figurines will be inside, and so it is a matter of chance whether they get the figurines that they want most. While the price is the same across all $4$-packs, the weight is not, as each figurine has a distinct integer weight in grams.

Before MTA releases its new line of figurines, its Chief Financial Officer (CFO) would like an executive summary with information about the proposed $4$-packs. Specifically, the CFO wants to know: the greatest possible weight of a $4$-pack, the smallest possible weight of a $4$-pack, the number of distinct weights that a $4$-pack could have, and the expected weight of a $4$-pack. Note that the expected weight is the average (mean) weight across all possible distinct $4$-packs, where two $4$-packs are distinct if and only if one $4$-pack has a different number of figurines of any particular weight than the other $4$-pack. So, for example, a $4$-pack with weights $\{ 2,2,3,5\} $ is distinct from a $4$-pack with weights $\{ 2,3,3,4\} $ (even though they both have the same total weight). Also, a $4$-pack with weights $\{ 2,2,3,5\} $ is distinct from a 4-pack with weights $\{ 2,3,3,5\} $.

-----Input-----
The input consists of a single test case. The first line of the input contains a single integer $N$, the number of different figurines to be produced, where $1\leq N\leq 40000$. The next line contains $N$ space-separated integers representing the weight in grams of each of the figurines. Each weight, $k$, is an integer satisfying $1\leq k\leq 60000$, and all $N$ weights are distinct.

-----Output-----
The output should consist of a single line with $4$ space-separated values. The first value should be an integer, the maximum weight of a $4$-pack in grams for the given set of figurines. The second value should be an integer, the minimum weight of a $4$-pack in grams for the given set of figurines. The third value should be an integer, the number of distinct weights that the $4$-packs can have for the given set of figurines. The fourth value should be a floating-point value, the expected weight of a $4$-pack in grams for the given set of figurines, with an absolute or relative error of at most $10^{-4}$.

-----Examples-----
Sample Input:
4
1 2 4 7
Sample Output:
28 4 21 14.0
user: Alice and Bob are big fans of math. In particular, they are very excited about playing games that are related to numbers. Whenever they see a puzzle like Sudoku, they cannot stop themselves from solving it. The objective of Sudoku is to fill a $9\times 9$ grid with digits so that each column, each row, and each of the nine ($3\times 3$) subgrids that compose the grid (also called “boxes”, “blocks”, or “regions”) contains all of the digits from $1$ to $9$. The puzzle setter provides a partially completed grid, which for a well-posed puzzle has a single solution.

After many years of solving Sudoku problems, Alice and Bob are tired of Sudoku. They have been trying to develop a harder variation of Sudoku, which they are calling Superdoku. In Superdoku, the grid is bigger – $n\times n$ instead of just $9\times 9$. However, the “block” constraints are impossible to formulate when there are no further constraints on $n$. Therefore, there are no block constraints in Superdoku. Instead, the goal is simply to make sure that each column and each row in the grid contains all of the integers from $1$ to $n$. After playing for a while in the standard way (where any of the grid cells may have previously been filled in), they decide that the game is too difficult and they want to simplify it. Therefore, they decide to make the initial grid further constrained. They constrain the board by filling in the first $k$ rows completely.

Alice and Bob both believe that Superdoku is solvable. However, since $n$ could be very big, it may still take a long time to figure out a solution. They don’t want to spend too much time on this single game, so they are asking for your help!

-----Input-----
The input consists of a single test case. The first line lists two space-separated integers $1 \le n \le 100$ and $0 \le k \le n$, denoting the size of the grid $(n\times n)$ and the number of rows $k$ that are already filled in. Each of the following $k$ lines contains $n$ space-separated integers, denoting the first $k$ given rows. All integers in these $k$ lines are between $1$ and $n$.

-----Output-----
Output either “yes” or “no” on the first line, indicating if there is a solution. If there is no solution, do not output anything more. If there is a solution, output $n$ more lines, each containing $n$ space-separated integers, representing a solution. If there are multiple solutions, output any one of them.

-----Examples-----
Sample Input:
4 2
1 2 3 4
2 3 4 1
Sample Output:
yes
1 2 3 4 
2 3 4 1 
3 4 1 2 
4 1 2 3
user: The Martingale casino is creating new games to lure in new gamblers who tire of the standard fare. Their latest invention is a fast-paced game of chance called Stop Counting!, where a single customer plays with a dealer who has a deck of cards. Each card has some integer value.

One by one, the dealer reveals the cards in the deck in order, and keeps track of the sum of the played cards and the number of cards shown. At some point before a card is dealt, the player can call “Stop Counting!” After this, the dealer continues displaying cards in order, but does not include them in the running sums. At some point after calling “Stop Counting!”, and just before another card is dealt, the player can also call “Start Counting!” and the dealer then includes subsequent cards in the totals. The player can only call “Stop Counting!” and “Start Counting!” at most once each, and they must call “Stop Counting!” before they can call “Start Counting!”. A card is “counted” if it is dealt before the player calls “Stop Counting!” or is dealt after the player calls “Start Counting!”

The payout of the game is then the average value of the counted cards. That is, it is the sum of the counted cards divided by the number of counted cards. If there are no counted cards, the payout is $0$.

You have an ‘in’ with the dealer, and you know the full deck in order ahead of time. What is the maximum payout you can achieve?

-----Input-----
The first line of the input contains a single integer $1 \leq N \leq 1000000$, the number of cards in the deck.

The second line of input contains $N$ space-separated integers, the values on the cards. The value of each card is in the range $[-10^{9}, 10^{9}]$. The cards are dealt in the same order they are given in the input.

-----Output-----
Output the largest attainable payout. The answer is considered correct if the absolute error is less than $10^{-6}$, or the relative error is less than $10^{-9}$.

-----Sample Explanation-----
In the first sample, by calling “Stop Counting!” before the $-10$ and “Start Counting!” before the final $10$, we can achieve an average of $10.0$ with the cards that are counted.

In the second sample, all values are negative, so the best strategy is to call “Stop Counting!” before the first card is dealt and call “Start Counting!” after the last card is dealt. Since no cards were counted, the average of the counted cards is $0.0$.

-----Examples-----
Sample Input:
5
10 10 -10 -4 10
Sample Output:
10.000000000
user: Ms. Hall wants to teach her class about common factors. She arranges her students in a circle and assigns each student an integer in the range $[2,10^9]$. She also provides the students with crepe paper streamers. The students are to stretch these streamers between pairs of students, and pull them tight. But, there are some rules.
 - Two students can stretch a streamer between them if and only if their assigned integers share a factor other than $1$.
 - There is exactly one path, going from streamer to streamer, between any two students in the circle.
 - No streamers may cross.
 - Any given student may hold an end of arbitrarily many streamers.

Suppose Ms. Hall has four students, and she gives them the numbers $2$, $3$, $30$ and $45$. In this arrangement, there is one way to stretch the streamers:

In this arrangement, there are three ways to stretch the streamers:  

In how many ways can the students hold the streamers subject to Ms. Hall’s rules? Two ways are different if and only if there is a streamer between two given students one way, but none between those two students the other way.

-----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 $n$ ($2 \le n \le 300$), which is the number of Ms. Hall’s students.

Each of the next $n$ lines will hold an integer $x$ ($2 \le x \le 10^9$). These are the numbers held by the students, in order. Remember, the students are standing in a circle, so the last student is adjacent to the first student.

-----Output-----
Output a single integer, which is the number of ways Ms. Hall’s students can satisfy her rules. Since this number may be very large, output it modulo $10^9+7$.

-----Examples-----
Sample Input:
4
30
3
2
45
Sample Output:
1