meetforever/GRPO_CL huggingface dataset:
user: A bicycle race is being organized in a land far, far away. There are $N$ towns in the land, numbered $1$ through $N$. There are also $M$ one-way roads between the towns. The race will start in town $1$ and end in town $2$.

How many different ways can the route be set? Two routes are considered different if they do not use the exact same roads.

-----Input-----
The first line of input contains two integers $N$ and $M$ ($1 \le N \le 10000$, $1 \le M \le 100000$), the number of towns and roads.

Each of the next $M$ lines contains two different integers $A$ and $B$, representing a road from town $A$ to town $B$.

Towns may be connected by more than one road.

-----Output-----
Output the number of distinct routes that can be set on a single line. If that number has more than nine digits, output only the last nine digits of the number. If there are infinitely many routes, output “inf”.

-----Examples-----
Sample Input 1:
6 7
1 3
1 4
3 2
4 2
5 6
6 5
3 4
Sample Output 1:
3

Sample Input 2:
6 8
1 3
1 4
3 2
4 2
5 6
6 5
3 4
4 3
Sample Output 2:
inf
user: Organising a group trip for the elderly can be a daunting task... Not least because of the fussy participants, each of whom will only make the trip on condition that some other participant also comes.

After some effort, you have taken from each of your participants a number, indicating that this participant will refuse to join the excursion unless the participant with that number also joins– the less choosy simply give their own number. This would be easy enough to resolve (just send all of them) but the bus you are going to use during the trip has only a fixed number of places.

-----Task-----
Given the preferences of all participants, find the maximum number of participants that can join.

-----Input-----
The first line of input contains two integers $n$ and $k$ ($1 \leq k \leq n \leq 1000$), where $n$ denotes the total number of participants and $k$ denotes the number of places on the bus.

The second line contains $n$ integers $x_{i}$ for $i=1,2,\ldots ,n$, where $1 \leq x_ i \leq n$. The meaning of $x_ i$ is that the $i$-th participant will refuse to join the excursion unless the $x_ i$-th participant also joins.

-----Output-----
Output one integer: the maximum number of participants that can join the excursion, so that all the participants’ preferences are obeyed and the capacity of the bus is not exceeded.

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

Sample Input 2:
12 3
2 3 4 5 6 7 4 7 8 8 12 12
Sample Output 2:
2
user: It is now far into the future and human civilization is ancient history. Archaeologists from a distant planet have recently discovered Earth. Among many other things, they want to decipher the English language.

They have collected many printed documents to form a dictionary, but are aware that sometimes words are not spelled correctly (typos are a universal problem). They want to classify each word in the dictionary as either correct or a typo. Naïvely, they do this using a simple rule: a typo is any word in the dictionary such that deleting a single character from that word produces another word in the dictionary.

Help these alien archaeologists out! Given a dictionary of words, determine which words are typos. That is, which words result in another word in the dictionary after deleting a single character.

For example if our dictionary is $\{ \texttt{hoose}, \texttt{hose}, \texttt{nose}, \texttt{noises}\} $. Then hoose is a typo because we can obtain hose by deleting a single ’o’ from hoose. But noises is not a typo because deleting any single character does not result in another word in the dictionary.

However, if our dictionary is $\{ \texttt{hoose}, \texttt{hose}, \texttt{nose}, \texttt{noises}, \texttt{noise}\} $ then the typos are hoose, noises, and noise.

-----Input-----
The first line of input contains a single integer $n$, indicating the number of words in the dictionary.

The next $n$ lines describe the dictionary. The $i^\textrm {th}$ of which contains the $i^\textrm {th}$ word in the dictionary. Each word consists only of lowercase English letters. All words are unique.

The total length of all strings is at most $1000000$.

-----Output-----
Display the words that are typos in the dictionary. These should be output in the same order they appear in the input. If there are no typos, simply display the phrase NO TYPOS.

-----Examples-----
Sample Input 1:
5
hoose
hose
nose
noises
noise
Sample Output 1:
hoose
noises
noise

Sample Input 2:
4
hose
hoose
oose
moose
Sample Output 2:
hoose
moose
user: Mirko decided to open a new business—bank vaults. A branch of the bank can be visualized in a plane, vaults being points in the plane. Mirko’s branch contains exactly $L\cdot (A+1+B)$ vaults, so that each point with integer coordinates inside the rectangle with corners $(1, -A)$ and $(L, B)$ contains one vault.

The vaults are watched by two guards—one at $(0, -A)$, the other at $(0, B)$. A guard can see a vault if there are no other vaults on the line segment connecting them.

A vault is not secure if neither guard can see it, secure if only one guard can see it and super-secure if both guards can see it. Given $A$, $B$ and $L$, output the number of insecure, secure and super-secure vaults.

-----Input-----
The first line contains integers $A$ and $B$ separated by a space ($1 \leq A \leq 2000, 1 \leq B \leq 2000$). The second line contains the integer $L$ ($1 \leq L \leq 1000000000$).

-----Output-----
Output on three separate lines the numbers of insecure, secure and super-secure vaults.

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

Sample Input 2:
2 3
4
Sample Output 2:
0
16
8
user: You are at home and about to drive to work. The road you will take is a straight line with no speed limit. There are, however, traffic lights precisely every kilometer, and you can not pass a red light. The lights change instantaneously between green and red, and you can pass a light whenever it is green. You can also pass through a light at the exact moment of changing colour. There are no traffic lights at the start or the end of the road.

Now your car is special; it is an iCar, the first Orange car, and it has only one button. When you hold down the button, the car accelerates at a constant rate of $1 \mathrm{m}/\mathrm{s}^2$; when you release the button the car stops on the spot.

You have driven to work many times, so you happen to know the schedules of the traffic lights. Now the question is, how quickly can you get to work?

-----Input-----
The first line contains a single integer $n$, the length of the road in kilometers ($1 \leq n \leq 16$). Each of the next $n-1$ lines contains $3$ integers $t_ i$, $g_ i$ and $r_ i$, the first time the $i$-th light will switch from red to green after the moment you start driving the car; the green light duration, and the red light duration ($40\leq g_ i,r_ i\leq 50$; $0 \leq t_ i < g_ i+r_ i$). Times are given in seconds.

You may assume that any light with $t_ i > r_ i$ is green at the time you start driving the car, and switches to red $t_ i - r_ i$ seconds later.

-----Output-----
Output the minimum time required to reach the end of the road. Answers within a relative or absolute error of $10^{-6}$ will be accepted.

-----Examples-----
Sample Input 1:
1
Sample Output 1:
44.72135955

Sample Input 2:
2
50 45 45
Sample Output 2:
68.52419365
user: You are given an array of integers of length $N$. Let $s_1, s_2, \ldots , s_ q$ be the lexicographically sorted array of all its non-empty subsequences. A subsequence of the array is an array obtained by removing zero or more elements from the initial array. Notice that some subsequences can be equal and that it holds $q = 2^N - 1$.

An array $A$ is lexicographically smaller than array $B$ if $A_ i < B_ i$ where $i$ is the first position at which the arrays differ, or if $A$ is a strict prefix of array $B$.

Let us define the hash of an array that consists of values $v_1, v_2, \ldots , v_ p$ as:\[ h(s) = (v_1 \cdot B^{p-1} + v_2 \cdot B^{p-2} + \ldots + v_{p-1} \cdot B + v_ p ) \bmod M \]

where $B$, $M$ are given integers.

Calculate $h(s_1) h(s_2), \ldots , h(s_ K)$ for a given $K$.

-----Input-----
The first line contains integers $N$, $K$, $B$, $M$ ($1 \leq N \leq 100000$, $1 \leq K \leq 100000$, $1 \leq B, M \leq 1000000$). The second line contains integers $a_1, a_2, a_3, \ldots , a_ N$ ($1 \leq a_ i \leq 100000$). In all test cases, it will hold $K \leq 2^N - 1$.

-----Output-----
Output $K$ lines, the $j$-th line containing $h(s_ j)$.

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

Sample Input 2:
3 4 2 3
1 3 1
Sample Output 2:
1
1
0
2
user: One day Mirko, while he was walking through the high grass, stumbled upon a sequence of $N$ colored marbles. Soon he noticed that if he touches $K$ or more consecutive marbles of the same color, they start to twinkle and then he could wish them to magically vanish, although he doesn’t have to do that immediately (see $3$rd sample). Fortunately, Mirko brought an inexhaustible supply of marbles from home, so he can insert a marble of any color anywhere in the array (at the beginning, between any two existing marbles, or at the end). Help Mirko find the smallest number of marbles he must insert into the sequence before he could make all of the marbles vanish.

-----Input-----
The first line of input contains two integers $N$ ($1 \leq N \leq 100$) and $K$ ($2 \leq K \leq 5$) – the number of marbles in the initial sequence and the minimal number of consecutive marbles of the same color he could wish to vanish. The next line contains exactly $N$ integers between $1$ and $100$ (inclusive), separated by one space. Those numbers represent colors of marbles in the sequence Mirko found.

-----Output-----
The output should contain only one line with a single integer number – the minimal number of marbles Mirko has to insert to achive the desired effect.

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

Sample Input 2:
5 3
2 2 3 2 2
Sample Output 2:
2
user: Everyone knows of the secret agent double-oh-seven, the popular Bond (James Bond). A lesser known fact is that he actually did not perform most of his missions by himself; they were instead done by his cousins, Jimmy Bonds. Bond (James Bond) has grown weary of having to distribute assign missions to Jimmy Bonds every time he gets new missions so he has asked you to help him out.

Every month Bond (James Bond) receives a list of missions. Using his detailed intelligence from past missions, for every mission and for every Jimmy Bond he calculates the probability of that particular mission being successfully completed by that particular Jimmy Bond. Your program should process that data and find the arrangement that will result in the greatest probability that all missions are completed successfully. Note that the number of missions is equal to the number of Bonds, so that every Jimmy Bond needs to be assigned a single mission each.

Note: the probability of all missions being completed successfully is equal to the product of the probabilities of the single missions being completed successfully.

-----Input-----
The first line will contain an integer $N$, the number of Jimmy Bonds and missions ($1 \le N \le 20$).

The following $N$ lines will contain $N$ integers between $0$ and $100$, inclusive. The $j$:th integer on the $i$:th line is the probability that Jimmy Bond $i$ would successfully complete mission $j$, given as a percentage.

-----Output-----
Output the maximum probability of Jimmy Bonds successfully completing all the missions, as a percentage.

Your answer should have an absolute error of at most $10^{-6}$.

-----Explanation of third sample-----
If Jimmy bond $1$ is assigned the $3$:rd mission, Jimmy Bond $2$ the $1$:st mission and Jimmy Bond $3$ the $2$:nd mission the probability is: $1.0 \cdot 0.13 \cdot 0.7 = 0.091 = 9.1\% $. All other arrangements give a smaller probability of success.

-----Examples-----
Sample Input 1:
2
100 100
50 50
Sample Output 1:
50

Sample Input 2:
2
0 50
50 0
Sample Output 2:
25
user: In Delft there are a number of bridges that are still being operated by a human, known as the bridge operator. One such bridge operator will soon retire, hence there is the need for a replacement. The Bridge And Poker Committee has decided to use a computer program to automatically open and close the bridge, eliminating the need for human interaction.

However, the computer program still needs to be written. The requirements for this project are as follows:
 - No boat may be forced to wait for more than $30$ minutes.
 - The amount of time during which the bridge is unavailable to road traffic must be as small as possible while still satisfying requirement 1.

It takes $60$ seconds to raise or lower the bridge. During this time the bridge is not available to either road traffic or water traffic.

Boats arrive at the bridge at predictable times. It takes $20$ seconds for a boat to sail through the bridge, assuming the bridge is already fully raised.

If the bridge is not fully raised when a boat arrives, the boat must wait. If there are boats waiting when the bridge becomes fully raised, these boats pass through the bridge one-by-one, which takes $20$ seconds per boat. The bridge must remain fully raised as long as there are still boats sailing through! As soon as all boats have passed, the bridge may be lowered. But it might be more efficient to keep the bridge raised for a little while longer if the next boat is soon to arrive.

Given the arrival times of all boats, operate the bridge such that all boats can pass through without any boat waiting longer than $30$ minutes. What is the total amount of time during which the bridge is unavailable to road traffic?

-----Input-----
The first line contains an integer $N$, the number of boats that must pass the bridge ($1 \leq N \leq 4000$).

Then follow $N$ lines, each containing an integer $T_ i$, the time at which boat $i$ will arrive at the bridge in seconds ($60 \leq T_ i \leq 10^5$).

Boats are sorted by increasing time of arrival, and never arrive within $20$ seconds of each other ($i < j$ implies $T_ i + 20 \leq T_ j$).

-----Output-----
Write one line with an integer, the total number of seconds during which the bridge must be unavailable for road traffic in order for all boats to pass the bridge.

-----Examples-----
Sample Input 1:
2
100
200
Sample Output 1:
160

Sample Input 2:
3
100
200
2010
Sample Output 2:
250
user: You and your friends have just returned from a beautiful vacation in the mountains of the Netherlands. When on vacation, it’s annoying to split the bill on every expense every time, so you just kept all the receipts from the vacation, and wrote down who paid how much for who. Now, it is time to settle the bill. 

You could each take all the receipts showing that someone paid something for you, and then pay that person back. But then you would need a lot of transactions, and you want to keep up the lazy spirit from your trip. In the end, it does not matter who transfers money to whom; as long as in the end, everyone’s balance is $0$.

Can you figure out the least number of transactions needed to settle the score? Assume everyone has enough spare cash to transfer an arbitrary amount of money to another person.

-----Input-----
Input consists of:
 - A line containing two integers $M$, the number of people in the group, with $1\leq M \leq 20$, and $N$, the number of receipts from the trip, with $0\leq N\leq 1000$.
 - $N$ lines, each with three integers $a, b, p$, where $0 \leq a,b < M$, and $1 \leq p \leq 1000$, signifying a receipt showing that person $a$ paid $p$ euros for person $b$.

-----Output-----
Output a single line containing a single integer, the least number of transactions necessary to settle all bills.

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

Sample Input 2:
5 5
0 1 3
1 2 3
2 3 3
3 4 3
4 0 3
Sample Output 2:
0
user: Zara, an aspiring architect and urban planner, has drawn out what she considers to be the perfect skyline. As Zara is still aspiring she must use her young daughter, Pippa, to test out her designs. In order to test out the designs Pippa must build them out of her building blocks! The building blocks Pippa has have a uniform width and depth, but come in different heights $h$. Zara’s description for Pippa will consist of a list of buildings, each with a target height $b$.

Pippa must then use some (not necessarily all) of her blocks to stack together such that the sum of the heights of the blocks in each stack corresponds to the height of the building in the skyline. Since Pippa prefers building instead of solving puzzles she wants you to determine how she must stack the blocks so that she must only do the stacking!

-----Input-----
The input consists of a single test case. The first line of this test case contains two integers $N$ and $S$ ($1 \le N \le 15$ and $1 \le S \le 15$), where $N$ is the number of blocks Pippa has and $S$ is the number of buildings in the skyline Zara made.

The next line contains $N$ integers ($1 \le h_ i \le 10^{9}$) representing the heights of each of the blocks. The last line contains $S$ integers ($1 \le b_ i \le 10^{9}$) representing the heights of each of the buildings.

-----Output-----
If it is possible for Pippa to build Zara’s skyline then output $S$ lines. On each line output a single number $s_ i$ representing the number of blocks needed to build building $i$ where $i$ corresponds to the $i^{\text {th}}$ building listed in the input. This should be followed (on the same line) by $s_ i$ numbers $j$ representing the blocks of the input used in building $i$, where $j$ represents the $j^{\text {th}}$ block appearing the input.

If no combination of the blocks can build the desired skyline then output -1.

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

Sample Input 2:
4 2
3 3 2 2
6 3
Sample Output 2:
-1
user: On the forest-filled moon of Endor there is, if we are to believe the Guinness Book of Records, the longest stick in the whole galaxy. On that stick of $L$ meters in length there are $N$ cheerful chameleons. Each chameleon is moving along the stick with constant speed of $1\mathrm{m}/\mathrm{s}$ in one of two possible directions (left or right) and is colored in one of the possible $K$ colors.

It is known that the chameleons of Endor worship the ancient ant laws that dictate that the walk along the stick must be continued until the end of the stick is reached (which means getting off it), and when a collision with another chameleon takes place, you must turn $180$ degrees and continue the walk in the opposite direction. Additionally, after a chameleon colored in colour $a$ moving to the left collides with a chameleon colored in colour $b$ moving to the right, the chameleon moving to the left before the collision takes the color of the chameleon moving to the right before the collision (so, color $b$), while the chameleon moving to the right before the collision takes a new color $(a + b) \bmod K$.

If you are given the initial positions, colors and directions of movement of all the chameleons, for each color determine the total trip taken by the chameleons in that color before getting off the stick.

-----Input-----
The first line of input contains the integers $N$, $K$ and $L$ ($1 \leq N \leq 100000$, $1 \leq K \leq 40$, $1 \leq L \leq 1000000$) from the task. The $i$-th of the following $N$ lines contains the integer $d_ i$ ($0 \leq d_ i \leq L$) that denotes the distance between the $i$-th chameleon and the left end of the stick, then the integer $b_ i$ ($0 \leq b_ i \leq K - 1$) that denotes the color of the $i$-th chameleon and the character ‘L’ (left) or ‘D’ (right) that denote the direction of movement of the $i$-th chameleon. All integers $d_ i$ will be unique and given in strictly ascending order.

-----Output-----
The output must contain $K$ lines, the $i$-th line containing the total trip taken by the chameleons in color $i$.

-----Examples-----
Sample Input 1:
2 3 10
0 0 D
10 1 L
Sample Output 1:
10.0
10.0
0.0

Sample Input 2:
4 3 7
1 0 D
3 0 D
4 1 L
6 2 D
Sample Output 2:
10.0
4.0
1.0
user: Rookie Pirate Captain Jack Albatross has secretly laid anchor in a hidden bay not far from Port Wine, the local meeting point for grizzled pirates and denizens of the sea. Disguising as a messenger, you have infiltrated the service of the dreaded legendary Captain Stevie, and you are entrusted with carrying small encrypted messages between Captain Stevie and his staff of shipmates. Your task is to break the encryption of the messages in aid of young Captain Albatross, giving him a much needed edge in his quest for peg-legged fame. Fortunately, you (in contrast to Captain Albatross) have been around for a while and you’ve come to know what words to expect in written pirate messages. The following is a table of words having common occurrences:

“be”

“our”

“rum”

“will”

“dead”

“hook”

“ship”

“blood”

“sable”

“avenge”

“parrot”

“captain”

In a (mixed alphabet) substitution cipher, each letter of the plain text is replaced with another, in this case arbitrary but unique, letter from the alphabet. Given a text encrypted with a substitution cipher and the given set of the twelve known words above, your task is to decode the encrypted text and provide us and Captain Albatross with the plain text. This can be accomplished if and only if there is a unique substitution mapping from the encrypted letters to plain text letters such that
 - a subset $S$ of the twelve words show up in the plain text, and
 - the number of different letters in the words in $S$ is equal to the number of different letters in the encrypted text.

Note that not all the known words may be present in the plain text and that replacements are not mutual (’a’ being encrypted by ’h’ does not necessarily mean that ’h’ is encrypted by ’a’).

-----Input-----
The input contains the encrypted text on one line. The text contains at most 200 characters from the set ’a’, …, ’z’, ’ ’. Only the letters ’a’, …, ’z’ of the plain text are encrypted, the spaces are kept unencrypted.

-----Output-----
The decrypted plain text if possible, or the string “Impossible” if the text cannot be uniquely decrypted using the set of known words.

-----Examples-----
Sample Input 1:
ex eoii jpxbmx cvz uxju sjzzcn jzz

Sample Output 1:
we will avenge our dead parrot arr

Sample Input 2:
wl jkd

Sample Output 2:
Impossible