meetforever/GRPO_CL huggingface dataset:
user: loglogloglog. Brandon Greg Jr. is sometimes a drowning combinatorist.

A permutation of length $n$ can be represented as some order of the integers $\{ 1,2,\ldots ,n\} $. A run of length $k$ in a permutation is $k$ consecutive elements that appear in increasing or decreasing order. For example, $[5,2,4,3,1]$ is a permutation of length $5$, where (among other runs), $[2,4]$ is a run of length $2$ and $[4,3,1]$ is a run of length $3$.

Brandon found a way to count the number of permutations of length $n$ that have runs of length at most $k$. He wants you to do the same so he could check his answers. Since the numbers get very large, Brandon will ask for it modulo some prime $p$ of his choice.

-----Input-----
The only line of input contains three space-separated integers $n, k, p$ ($1\le n\le 2000$, $2\le k\le 7$, $10^8<p\le 10^9+9$), where $n$ is the length of the permutations, $k$ is the maximum length of a run, and $p$ is the prime of Brandon’s choice.

-----Output-----
Output one integer representing the number of permutations of length $n$ that have runs of length at most $k$, modulo $p$.

-----Examples-----
Sample Input 1:
1 7 1000000007
Sample Output 1:
1

Sample Input 2:
3 2 1000000007
Sample Output 2:
4
user: Luka found a very unusual game board in his attic. Surprisingly, it consists of $R \cdot C$ square cells. The rows are numbered $0$ to $R-1$ top to bottom and the columns $0$ to $C-1$ left to right.

What makes the board unusual is the way in which the cells are coloured. Each cell is either grey or white:
 - white, if the row and column numbers of the cell, when represented in binary, have at least one digit 1 in the same position. For example, the cell $(4, 5)$ would be white.
 - grey, otherwise. For example, the cell $(2, 5)$ would be grey.

The following image shows a board of size 10·10. 

Luka’s hedgehog likes walking on this unusual board and does it in an unusual way. The hedgehog starts his walk in the cell $(0, 0)$ and continues in the zig-zag pattern as in the second image above. While the hedgehog is walking, Luka counts how many grey squares it visited.

After visiting $K$ squares, the hedgehog gets tired and falls asleep. Luka then goes to bed too, happy that he was able count the grey squares.

Knowing the dimensions of the board and the number $K$ beforehand, however, it is possible to write a program that calculates the result faster. This is your task.

-----Input-----
The first line contains two integers $R$ ($1 \le R \le 1000000$) and $C$ ($1 \le C \le 1000000$), the dimensions of the board.

The second line contains the integer $K$ ($1 \le K \le R \cdot C$), the total number of squares the hedgehog visits.

Note that this number may not fit in a 32-bit integer.

-----Output-----
Output the number of grey cells the hedgehog visits.

-----Examples-----
Sample Input 1:
10 10
6
Sample Output 1:
5

Sample Input 2:
3 5
11
Sample Output 2:
8
user: You run a delivery company and must deploy a fleet of vehicles to deliver goods to clients. All of the goods and delivery trucks are initially located at your warehouse. 

The road network consists of one-way streets between junctions. The warehouse and clients are all located at a junction. You know the driving time across each street.

You guarantee extremely fast shipping: the trucks start driving immediately at the start of the day and each client $i$ will receive the package at time $T_ i$ where $T_ i$ is the shortest possible driving time for a truck to go from the warehouse to the location of the client $i$.

What is the minimum number of trucks you have to deploy to ensure this guarantee is met? That is, what is the minimum number of trucks such that it is possible to give each truck a driving route so that every client $i$ is visited by some truck at time $T_ i$. Assume it takes no time to load the trucks with the appropriate goods at the start of the day, and it takes no time to drop goods off at a client once the truck arrives at the client. These goods are small enough that each truck can carry goods for as many clients as necessary.

-----Input-----
The input consists of a single test case. The first line of each test case consists of three numbers $N$, $M$, and $C$. Here $N$ denotes the number of junctions in the road network ($2 \leq N \leq 10^3$), $M$ denotes the number of streets ($1 \leq M \leq 10^5$), and $C$ denotes the number of clients ($1 \leq C \leq 300$, $C < N$).

The junctions are numbered $0$ to $N-1$. The warehouse is always at junction $0$. The second line consists of $C$ distinct integers between $1$ and $N-1$ indicating the junctions where the clients reside.

The rest of the input consists of $M$ lines, each containing integers $U,V,W$ where $0 \leq U,V \leq N-1$ and $U \neq V$. This indicates there is a one-way street from $U$ to $V$ with driving time $W$. Each street’s driving time $W$ satisfies $1 \leq W \leq 10^9$. It will always be possible to reach every client from the warehouse.

There will be at most one street from a vertex $U$ to another vertex $V$, but there may be streets from both $U$ to $V$ and from $V$ to $U$.

-----Output-----
Output a single integer that is the minimum number of vehicles required to ensure each client $i$ is visited at time $T_ i$ by some vehicle.

-----Explanations of Sample Inputs-----
In the first sample, one vehicle can follow the path $(0,1,2)$ and the other can follow $(0,3)$. In the second sample, the only solution is to use paths $(0,1)$, $(0,2)$, and $(0,3)$. In the final sample, one vehicle can follow $(0,1)$, another $(0,4,6)$, and the last one $(0,2,3,5,7)$.

-----Examples-----
Sample Input 1:
4 5 3
1 2 3
0 1 1
0 3 1
0 2 2
1 2 1
3 2 1
Sample Output 1:
2

Sample Input 2:
4 5 3
1 2 3
0 1 1
0 3 1
0 2 1
1 2 1
3 2 1
Sample Output 2:
3
user: Fluttershy has been put in charge of a high-end boutique in Manehattan, and she needs to deal with the customers that come in.

It is well-known that Fluttershy is, well, very shy. So shy, in fact, that the only time she feels comfortable dealing with a customer is when she is wearing the same type of clothing as that customer! When she wears a particular type of clothing, she is able to assume the customer’s personality and make a sale. There are $M$ types of clothing customers can wear.

She knows that $N$ customers will come today. The $i^\text {th}$ customer wears clothing of type $C_ i$ and will arrive exactly $T_ i$ moments of time after the opening of the shop. The citizens of Manehattan are all snooty and expect nothing but the speediest service, so if Fluttershy is not ready to serve them the moment they arrive, they will immediately leave. On the other hoof, Fluttershy is a very efficient salespony; we can assume that Fluttershy can make a sale in zero time. (This also means that if multiple customers arrive at the same moment of time, Fluttershy can serve all of them.)

Originally, Fluttershy is not wearing any clothing. To wear some clothing, she needs to follow the following process:
 - To put on clothing of type $i$, she needs to first ensure that she is not wearing any clothing. Then, she goes to the dressing room and takes $P_ i$ moments of time to put on the clothing. (That is, if she starts to put on clothing at time $t$, she is ready by time $t + P_ i$.)
 - To remove clothing of type $i$, she goes to the dressing room and takes $R_ i$ moments of time to remove the clothing. (That is, if she starts to remove clothing at time $t$, she has removed it by time $t + R_ i$ and can immediately start to put on another clothing.)

What is the maximum number of ponies she can serve?

-----Input-----
The first line of input contains two integers, $N$ ($1 \leq N \leq 200000$) and $M$ ($1 \leq M \leq N$), the number of customers that will arrive today, and the number of different types of clothing customers can wear, respectively.

The second line of input contains $M$ integers, $P_1, P_2, \dots , P_ M$ ($1 \leq P_ i \leq 10^{18}$), the required moments of time to put on each type of clothing.

The third line of input contains $M$ integers, $R_1, R_2, \dots , R_ M$ ($1 \leq R_ i \leq 10^{18}$), the required moments of time to remove each type of clothing.

The next $N$ lines of input contain the descriptions of the customers. In particular, the $i^\text {th}$ of these lines contains two integers $C_ i$ ($1\leq C_ i \leq M$) and $T_ i$ ($1 \leq T_ i \leq 10^{18}$), denoting that the $i^\text {th}$ customer is of type $C_ i$ and will arrive exactly $T_ i$ moments of time after the opening of the shop.

It is guaranteed that for all positive integers $i < N$, $T_ i \leq T_{i+1}$, and for all positive integers $k \leq M$, there exists some $j$ such that $C_ j = k$.

-----Output-----
Output a single integer on a line by itself, the maximum number of customers Fluttershy can serve.

-----Examples-----
Sample Input 1:
4 3
10 20 30
5 5 10
2 20
1 30
1 32
3 120
Sample Output 1:
3

Sample Input 2:
3 1
10
10
1 10
1 10
1 10
Sample Output 2:
3
user: Mirko and Slavko are playing a game. Mirko’s turn is first and he chooses a non-empty set of pairs of numbers between $1$ and $N$ (inclusive) under the condition that the numbers that comprise a pair are mutually relatively prime. The numbers that comprise a pair must be different. For example, for $N = 5$, Mirko could have chosen the following set of pairs: $\{ \{ 1, 2\} , \{ 3, 4\} , \{ 2, 5\} , \{ 3, 5\} \} $.

Slavko’s turn is second and his goal is to find a partition for Mirko’s set of pairs. Mirko’s set of pairs has a partition if an integer $x$ from the set $\{ 2, 3, \ldots , N \} $ exists such that, for each pair $\{ a, b\} $, one of the following holds:
 - $a, b < x$
 - $a, b \geq x$

For example, a set of pairs $\{ \{ 1, 2\} , \{ 3, 4\} \} $ has a partition $x = 3$. If a partition exists, Slavko will surely find it.

Mirko wins if Slavko can’t find a partition for his set. Determine how many different sets of pairs exists that Mirko can initially choose and be sure of his victory. Given the fact that the total number of sets can be very large, output the number modulo $1000000000$.

-----Input-----
The first line of input contains the integer $N$ ($1 \leq $N$ \leq 20$).

-----Output-----
The first and only line of output must contain the required number.

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

Sample Input 2:
3
Sample Output 2:
5
user: Alice and Bob spend this summer planning next year’s summer vacation. They want to travel from the old city of A-town to the popular summer resort B-ville, but they haven’t yet decided which other places to visit en route. There are $N$ places to visit in total, numbered from $1$ to $N$. A-town is place number $1$ and B-ville is place number $N$. Some places are linked by one-directional connections; a pair of places can be linked by multiple such connections.

Alice and Bob are currently in A-town. Each day they will decide where to travel next by choosing uniformly at random among the connections leaving their current place. Once they reach B-ville, they stop.

Alice and Bob want to book their trip back on a day where they are in B-ville with $95\% $ probability. And they really mean exactly$95\% $ – no more, no less! There is a $10$ day window when train tickets are affordable; if their return trip does no occur within that window they might as well cancel the entire vacation.

You are given a positive integer $L$ and have to find a number $T$ with $L \leq T \leq L + 9$ such that the probability of Alice and Bob being in B-ville after $T$ days is exactly $95\% $. If there is more than one valid $T$, report smallest one.

-----Input-----
The first line contains two integers $N$ and $L$, with $2 \leq N \leq 100$ and $1 \leq L \leq 10^6$. The following $N$ lines each contain $N$ integers. The $j$th integer on the $i$th line contains the number $a_{ij}$ of one-directional connections from place $i$ to place $j$, where $0 \leq a_{ij} \leq 10^9$.

There are no connections from a place to itself. No connections leave B-ville (the $N$th place). All other places have at least one outgoing connection, so Alice and Bob will never be stuck.

-----Output-----
Print the integer $T$ as described above. If no solution exists, print $-1$.

-----Examples-----
Sample Input 1:
3 1
0 11 9
1 0 10
0 0 0
Sample Output 1:
2

Sample Input 2:
4 3
0 1 0 19
0 0 2 0
0 5 0 3
0 0 0 0
Sample Output 2:
-1
user: -----Input-----
The first line of input contains a line with four non-negative integers, $2 \le n \le 500$, $0 \le m \le 10000$, $0 \le s \le n-1$ and $0 \le t \le n-1$, separated by single spaces, where $n$ is the numbers of nodes in the graph, $m$ is the number of edges, $s$ is the source and $t$ is the sink ($s \ne t$). Nodes are numbered from $0$ to $n-1$. Then follow $m$ lines, each line consisting of three (space-separated) integers $u$, $v$ and $c$ indicating that there is an edge from $u$ to $v$ in the graph with capacity $1 \le c \le 10^{8}$.

-----Output-----
The output should begin with a line containing three integers $n$, $f$, and $m’$. $n$ is the number of nodes in the flow graph (same as in input), $f$ is the size of a maximum flow from node $s$ to node $t$ in the flow graph, and $m’$ is the number of edges used in the solution. You may assume that $f < 2^{31}$.

Then there should be $m’$ lines, each containing three integers $u$, $v$ and $x$, indicating that $x$ units of flow are transported from $u$ to $v$. $x$ must be greater than $0$ and at most as big as the capacity from $u$ to $v$. Each pair of vertices $(u, v)$ should be given at most once in the output.

-----Examples-----
Sample Input 1:
4 5 0 3
0 1 10
1 2 1
1 3 1
0 2 1
2 3 10
Sample Output 1:
4 3 5
0 1 2
0 2 1
1 2 1
1 3 1
2 3 2

Sample Input 2:
2 1 0 1
0 1 100000
Sample Output 2:
2 100000 1
0 1 100000
user: Mirko and Slavko are playing a chess like game. The game is played on a non-standard chess board sized $R$ rows by $C$ columns. Each player starts with some number of chess kings. In chess kings can move from their current field to any of the $8$ neighbouring fields. Player spread is defined as the complete sum of distances between all pairs of pieces of the given player. The distance between two pieces is the smallest number of moves required for both pieces to reach the same field. No actual moves are performed when calculating the distance and as such enemy pieces do not influence the result. Mirko knows that the spread is a vital piece of strategic information and would like you to make him a program that will calculate both his and Slavko’s spread.

-----Input-----
The first line of input contains two integers $R$ and $C$ ($1 \le R , C \le 1000$), number of rows and columns. Next $R$ lines contain $C$ characters each. Character ‘M’ denotes Mirko’s piece, ‘S’ Slavko’s piece and ‘.’ denotes an empty field. There is at least one piece per player on the board. Otherwise the game would be over.

-----Output-----
In the first and only line of output you need to print exactly two integers. The first integers is the spread of Mirko’s and the second Slavko’s pieces.

-----Examples-----
Sample Input 1:
2 3
SMS
MMS
Sample Output 1:
3 5

Sample Input 2:
2 3
S.M
M..
Sample Output 2:
2 0
user: The distance between two integers is defined as the sum of the absolute result of subtracting their digits. For example, the distance between the numbers 4561 and 3278 is $|4 - 3| + |5 - 2| + |6 - 7| + |1 - 8| = 12$. If one of the numbers consists of fewer digits than the other, we fill it with leading zeroes. Therefore, the distance between the numbers 32 and 5678 is $|0 - 5| + |0 - 6| + |3 - 7| + |2 - 8| = 21$. You are given two integers $A$ and $B$. Calculate the sum of distances between each pair of numbers belonging in the interval $[A, B]$.

-----Input-----
The first and only line of input contains integers $A$, $B$ ($1 \leq A \leq B \leq 10^{50000}$).

-----Output-----
The first and only line of output must contain the required number from the text. Given that the number could be extremely large, output the answer modulo $1000000007$.

-----Examples-----
Sample Input 1:
1 5
Sample Output 1:
40

Sample Input 2:
288 291
Sample Output 2:
76
user: A Word Ladder is a puzzle in which you transform one word into another, by changing one letter at a time. But, there’s a catch: every word that you form in each step must be in the dictionary! Here’s an example of how to transform CAT into GAS:CAT $\rightarrow $ CAR $\rightarrow $ WAR $\rightarrow $ WAS $\rightarrow $ GAS

Of course, you want to use the fewest number of transitions possible. These puzzles can be tough, and often you’ll think to yourself: “Darn it! If only [some word] was in the dictionary!”

Well, now is your chance! Given a dictionary, and a starting and ending word, what ONE single word could you add to the dictionary to minimize the number of steps to get from the starting word to the ending word, changing only one letter at a time, and making sure that every word at every step is in the dictionary?

-----Input-----
Each input will consist of a single test case. Note that your program may be run multiple times on different inputs. Each test case will start with a line with a single integer $n$ ($2 \le n \le 1000$) which indicates the number of words in the dictionary. The dictionary will follow on the next $n$ lines, with one word per line. All words will consist of between $1$ and $8$ capital letters only, and all of the words in a test case will be of the same length. The first word in the list will be the starting word of the word ladder, and the second will be the ending word of the word ladder.

-----Output-----
Output exactly two lines. The first line holds the one single word that you would add to the dictionary, and the second holds an integer indicating the minimum number of steps to get from the starting word to the ending word, adding your word.

It is possible that there’s more than one word you can add that will make your path as short as possible. In this case, output the solution word that comes first alphabetically.

It is possible that there’s no word you can add that will that will make your path any shorter. In this case, output 0 (zero) as the word.

It is possible that there’s no word you can add that makes the solution possible. In this case, output 0 (zero) as the word, and -1 as the number of steps.

-----Examples-----
Sample Input 1:
3
CAT
DOG
COT
Sample Output 1:
COG
3

Sample Input 2:
2
CAT
DOG
Sample Output 2:
0
-1
user: Consider a sequence of $N$ integers where each integer between $1$ and $N$ appears exactly once.

A pair of numbers in the sequence is confused if the number that comes earlier in the sequence is larger than the later number.

The confusion of the sequence is the number of confused pairs in it. For example, the confusion of the sequence $(1, 4, 3, 2)$ is $3$ because there are $3$ confused pairs: $(4, 3)$, $(4, 2)$ and $(3, 2)$.

Write a program that calculates the number of sequences of length $N$ whose confusion is exactly $C$.

-----Input-----
The first and only line of input contains two integers, $N$ ($1 \le N \le 1000$) and $C$ ($0 \le C \le 10000$).

-----Output-----
Output the number of sequences modulo $1000000007$.

-----Examples-----
Sample Input 1:
10 1
Sample Output 1:
9

Sample Input 2:
4 3
Sample Output 2:
6
user: Hu the Tomb Raider has entered a new tomb! It is full of gargoyles, mirrors, and obstacles. There is a door, with treasure beyond. Hu must unlock the door guarding the treasure. On that door is written, in an ancient tongue, the secret to opening the door:Every face of every gargoyle shall see a face of a gargoyle.

This means that the gargoyles must be rotated in such a way that there is a path for a beam of light to connect each gargoyle’s face to another gargoyle’s face (possibly its own). The beam of light is reflected by mirrors.

The floorplan of the tomb can be described as a rectangular $n \! \times \! m$ grid of cells:
 - A dot (‘.’) represents an empty cell.
 - A hash (‘#’) represents an obstacle.
 - A slash (‘/’) represents a double-sided mirror, as does a Backslash (‘\’) .
 - A character ‘V’ represents a gargoyle with two faces facing top and bottom.
 - A character ‘H’ represents a gargoyle with two faces facing left and right.

In addition to the ‘\’ and ‘/’ mirrors, the tomb is surrounded by walls of mirrors. The following common sense about light is assumed:
 - Light travels in a straight line through empty cells.
 - Two beams of light can intersect without interfering with each other.
 - A ‘\’ mirror reflects light coming from the top/bottom/left/right to the right/left/bottom/top. A ‘/’ mirror reflects light coming from the top/bottom/left/right to the left/right/top/bottom.
 - Light is reflected by $180$ degrees when it hits a wall (walls are all mirrors).
 - Light is blocked by obstacles and gargoyles.

Hu may rotate any gargoyle by $90$ degrees. As time is running short, he wants to know the minimum number of gargoyles that have to be rotated in order to unlock the treasure door.

-----Input-----
The first line of input contains two space-separated integers $n$ and $m$ ($1 \leq n, m \leq 500$), which are the dimensions of the tomb.

Each of the next $n$ lines contains a string $s$ ($|s|=m$) with the characters described above. This is the floorplan of the tomb.

-----Output-----
Output a single integer, which is the minimum number of gargoyles that have to be rotated in order to unlock the treasure door. If the puzzle has no solution, output $-1$.

-----Sample Explanation-----


The above are illustrations of Sample Input/Output $1$ with the initial configuration on the left and the solution of the puzzle on the right. Three gargoyles are rotated to solve the puzzle.

-----Examples-----
Sample Input 1:
5 5
/.V.\
./.V.
..#..
.V.#.
\.V./
Sample Output 1:
3

Sample Input 2:
2 5
V...\
H...V
Sample Output 2:
-1
user: You have been captured by an evil organization which seeks to exploit your algorithm skills to improve their weapons. The first task they forced you to work on was to program guided torpedoes. Naturally, your goal is to instead make the torpedoes miss every ship whenever possible. 

A typical scenario where the torpedoes will be used is a 2D plane, where the torpedo is fired from $(0,0)$ towards $m$ ships in the positive $y$-direction. Every ship has the form of a line segment parallel to the $x$-axis, with integer coordinate endpoints. The torpedo is shaped like a single point, and every second it will travel from $(x,y)$ to either $(x-1,y+1)$, $(x,y+1)$, or $(x+1,y+1)$. In other words, it will always go one unit forward, but your program decides how much sideways it should go. If the torpedo hits one of the ships (including one of the endpoints of a ship) it will explode, destroying the ship. On the other hand, if the torpedo stays in one piece for $n$ seconds, its fuel will run out and it will harmlessly sink to the ocean floor.

Write a program which, given the position of the $m$ ships, finds out whether it is possible to avoid them all, and if so, outputs instructions to do it.

-----Input-----
The first line of input contains two integers $n$ and $m$ ($2 \leq n \leq 500000$ and $0 \leq m \leq 200000$), the number of seconds until the torpedo runs out of fuel, and the number of ships.

Then follow $m$ lines, each containing three integers $x_1, x_2, y$ ($-n \leq x_1 \leq x_2 \leq n$ and $1 \leq y < n$), indicating a ship with endpoints $(x_1, y)$ and $(x_2, y)$.

You may assume that no pair of ships touch each other.

-----Output-----
If it is possible to dodge all the ships, output a string of length $n$ containing the characters $-$, $0$, and $+$. This string represents how the torpedo should turn in each of the $n$ time steps. For example, if the first character is $+$, then the torpedo will start by going from $(0,0)$ to $(1,1)$. If there are multiple solutions, output any one of them. If it is impossible to avoid all the ships, output “impossible”.

-----Examples-----
Sample Input 1:
5 6
-3 -2 3
-2 -2 4
2 3 3
-1 1 2
0 1 4
2 5 1
Sample Output 1:
--+0-

Sample Input 2:
3 2
1 2 1
-2 0 2
Sample Output 2:
0+-