meetforever/GRPO_CL huggingface dataset:
user: Charles is traveling to the fantastic Fibonacci City. The city has $n$ mansions of various heights and $m$ bi-directional roads connecting the mansions. One of the most exciting tourist attraction in the city is the Fibonacci Tour, which is a simple path visiting a sequence of distinct mansions, so that every pair of consecutively visited mansions are directly connected by a road, and the heights of the mansions in the path form a continuous subsequence of the Fibonacci sequence:\[ 1, 1, 2, 3, 5, 8, \dots \]

Each number in the Fibonacci sequence equals the sum of the previous two numbers, except for the first two numbers which are defined to be one.

Charles will be happier if his tour is longer. What is the maximum length of a Fibonacci Tour he can take? The length of a tour is the number of mansions visited in the tour. Charles can start the tour at any mansion.

-----Input-----
The first line has two integers $n$ ($1\leq n\leq 10^5$) and $m$ ($0\leq m\leq 10^5$), giving the number of mansions and the number of roads in the city. The next line has $n$ positive integers no larger than $10^{18}$. The $i$-th integer gives the height of mansion $i$. The next $m$ lines each has two integers $a$ and $b$ ($1\leq a, b \leq n$, $a \neq b$), indicating that there is a bi-directional road between mansion $a$ and mansion $b$. Each pair of mansions is directly connected by at most one road.

-----Output-----
Output the length of the longest Fibonacci Tour Charles can take in the city. If there is no tour Charles can take, output zero.

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

Sample Input 2:
4 3
4 4 8 12
1 2
2 3
3 4
Sample Output 2:
1
user: In order to keep up with the rapid technological advancements, Mr. Panda needs to backup all the data on his computer to allow him to do a full upgrade of his computer. Since the upgrade will take some time, he will peruse the services of the company ACM (Association of Computer Maintenance) that provides storage servers for him to temporarily store his data.

Although ACM offers storage servers of any size, ACM wants to encourage more bulk purchases of servers of the same size as this saves cost. The cost of purchasing a set of storage servers that can each store $M$ bytes consists of a base cost of $M$ dollars plus $1$ dollar for every server of that size purchased.

Mr. Panda wants to store a total of $K$ bytes of data and in order to easily track the servers, he wants to purchase a set of storage servers of the same size to store the $K$ bytes of data using the least cost. However, Mr. Panda feels compelled not to waste any space. Thus, he requires that the total storage space of the servers purchased must be exactly $K$ bytes.

As storage space of technology increases exponentially with time, $K$ can be very large. You are given the prime factorization of $K$. Help Mr. Panda calculate the minimum cost required to store his data. As the minimum cost can be quite large, print the minimum cost modulo a prime number $10^9+7$.

-----Input-----
The input consists of one line with a string of even length consisting of at most $700$ representing the prime factorization of $K$. Every pair of two consecutive digits, starting from the first two digits, represents one prime factor. So input of length $N$ will have $N/2$ prime factors.

It is guaranteed that all the prime factors given are prime and that $K$ has at most $10^{10}$ divisors. Each prime factor will appear at most $100$ times.

-----Output-----
Output the minimum cost in one line as a single integer.Sample Data Explanation

In the first example, $K = 2 \cdot 3 \cdot 2 = 12$, so buy three servers with $M = 4$ bytes. Mr. Panda pays $4$ dollars base cost plus $3$ dollars for three servers purchased, or a total of $4+3 = 7$ dollars. Mr. Panda can also buy four servers with $M = 3$ bytes with the same cost of $7$ dollars.

In the second example, $K = 13 \cdot 11 = 143$, so buy eleven servers with $M = 13$ bytes. Mr. Panda pays $13$ dollars base cost plus $11$ dollars for eleven servers purchased, or a total $13+11 = 24$ dollars. Mr. Panda can also buy thirteen servers with $M = 11$ bytes with the same cost of $24$ dollars.

In the third example, $K = 11$, so buy one server with $M = 11$ bytes. Mr. Panda pays $11$ dollars base cost plus $1$ dollar for one server purchased, or a total $11+1 = 12$ dollars. Note that although he can get a cheaper cost if he were able to purchase storage servers of different sizes, he cannot do so.

-----Examples-----
Sample Input 1:
020302
Sample Output 1:
7

Sample Input 2:
1311
Sample Output 2:
24
user: Gold Leaf is a very thin layer of gold, with a paper backing. If the paper gets folded and then unfolded, the gold leaf will stick to itself more readily than it will stick to the paper, so there will be patches of gold and patches of exposed paper. Note that the gold leaf will always stick to itself, rather than the paper. In the following example, the paper was folded along the dashed line. Notice how the gold leaf always sticks to one side or the other, never both.

Consider a crude digital image of a sheet of gold leaf. If the area covered by a pixel is mostly gold, that will be represented by a ‘#’. If it’s mostly exposed paper, it will be represented by a ‘.’. Determine where the sheet was folded. The sheet was folded exactly once, along a horizontal, vertical, or 45 degree or 135 degree diagonal line. If the fold is horizontal or vertical, it is always between rows/columns. If the fold is diagonal, then the fold goes through a diagonal line of cells, and the cells along the fold are always ‘#’.

-----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 begin with a line with two integers, $n$ and $m$ ($2 \le n, m \le 25$), where $n$ is the number of rows, and $m$ is the number of columns of the image. Each of the next $n$ lines will contain exactly $m$ characters, all of which will be either ‘#’ or ‘.’. This represents a crudely collected digital image of the sheet of gold leaf. There is guaranteed to be at least one ‘.’, and there is guaranteed to be a solution.

-----Output-----
Output a single line with four integers, indicating the places where the fold hits the edges of the paper. Output them in this order:

    r1 c1 r2 c2

where $(r1,c1)$ and $(r2,c2)$ are row/column coordinates ($r$=row, $c$=column). The top left character of the image is $(1,1)$ and the bottom right is $(n,m)$.

If the fold is horizontal or diagonal, list the left coordinates before the right. If the fold is vertical, list the top coordinates before the bottom.

If the fold is horizontal, use the coordinates above the fold. If the fold is vertical, use the coordinates to the left of the fold. If the fold is diagonal, use the coordinates of the edge pixels that the fold goes through.

If more than one fold is possible, choose the one with the smallest first coordinate, then the smallest second coordinate, then third, then fourth.

-----Examples-----
Sample Input 1:
8 10
#.#..##..#
####..####
###.##....
...#..####
....##....
.#.##..##.
##########
##########
Sample Output 1:
3 1 3 10

Sample Input 2:
5 20
###########.#.#.#.#.
###########...#.###.
##########..##.#..##
###########..#.#.##.
###########.###...#.
Sample Output 2:
1 15 5 15
user: Pokemon Go has become a recent trend. Zapray becomes immediately addicted to the game after it comes out. Zapray’s Pokemon journey begins with a bag of $100$ Pokeballs. Each time he encounters a Pokemon, he keeps throwing Pokeballs to the Pokemon until he either catches the Pokemon or runs out of Pokeballs. Each Pokeball Zapray throws succeeds in catching the Pokemon with a constant probability $P$. If Zapray finds that he has no Pokeballs when he tries to catch a Pokemon, he gives up on that Pokemon (not catching it) and heads to the Pokeshop to buy a new bag of $100$ Pokeballs for $5$ dollars. Zapray moves on to search for his next Pokemon after he refills the Pokeballs. In particular, if Zapray successfully catches a Pokemon with his last Pokeball, he does not refill his Pokeballs until he encounters the next Pokemon and realizes that he has run out of Pokeballs.

Suppose Zapray encounters $N$ Pokemons in the next few days, what is the expected amount of money he would spend on Pokeballs?

-----Input-----
The input has an integer $N$ ($1\leq N\leq 10^9$) and a real number $P$ ($0 \leq P\leq 1$). $P$ is given with exactly three digits after the decimal point.

-----Output-----
Output the expected amount of money Zapray would spend on Pokeballs. Your answer is considered correct if it has an absolute or relative error of no more than $10^{-6}$ from the correct answer.

-----Note-----
Pokemon Go was developed by Niantic, Inc. Niantic does not endorse and has no involvement with the ProgNova contest.

-----Examples-----
Sample Input 1:
50 0.125
Sample Output 1:
16.339203308

Sample Input 2:
201 1.000
Sample Output 2:
5.000000000
user: Binary Town is holding its biennial elections. All $k$ positions on its council are open, and as usual Notnomde and Yraglac are the only ones running for each position.

Having long since abandoned quaint majority voting, Binary Town uses binary voting instead. In binary voting, all $v$ voters can cast any number of ballots, or not vote at all. The $k^\mathrm {th}$ least significant bits of the total number of ballots are then used to determine which candidate wins each position. That is, Notnomde wins position $j$ if the $j^\mathrm {th}$ least significant bit of the total number of ballots is $0$, otherwise Yraglac wins.

You know your neighbours well, and know exactly how many ballots $b_ i$ each one will cast if they choose to vote. For some reason, not everyone is happy with the electoral system so not everyone votes. Thankfully, you also know the probability $p_ i$ of each citizen voting.

Since you like Yraglac more than Notnomde, you want to cast the number of ballots which maximizes the expected number of positions held by Yraglac.

Suppose there are $k = 2$ positions and $v = 2$ voters, and you know with $50\% $ probability that the voter other than yourself will cast one ballot. If you cast one ballot, the total number of ballots could be $01_2$ or $10_2$ with equal probability, so the expected number of positions for Yraglac is $1$. If you cast two ballots, the total number of ballots could be $10_2$ or $11_2$ with equal probability, making Yraglac’s expected number of positions $1.5$. In this case, you then decide to cast two ballots since that maximizes Yraglac’s expected number of positions.

-----Input-----
The first line contains two space-separated integers $1 \leq k \leq 16$, the number of positions, and $2 \leq v \leq 100$, the number of voters (including yourself). It is followed by $v - 1$ lines, each containing two space-separated numbers: a decimal $0 \leq p_ i \leq 1$, the probability of voter $i$ casting ballots, and an integer $0 \leq b_ i \leq 2^k - 1$, the number of ballots voter $i$ would cast should they vote.

-----Output-----
Output on a single line the number of ballots $0 \leq b_ v \leq 2^k - 1$ you should cast to maximize the expected number of positions held by Yraglac. The answer is guaranteed to be unique.

-----Examples-----
Sample Input 1:
2 2
0.5 1
Sample Output 1:
2

Sample Input 2:
4 3
1 11
0.4 1
Sample Output 2:
3
user: Melita has just returned from the annual pig slaughter. Don’t worry, this is a regular thing in Croatia. The best part was the abundance of food! There was everything, starting from good spicy sausages, ham, black pudding, up to teewurst, top quality bacon and čvarci, all with warm white bread and butter. After these appetizers, it was the perfect time to whip up a deep pot full of sarma (Melita ate twentyish of them) as well as a large platter of fine roast pork, so soft that it almost melts in your mouth. They watered all of this down with copious gulps of the best dry white wine that made them even hungrier.

Butcher Bajs kept his award-winning ham for the very end. $N$ people attented the annual pig slaughter, labeled with numbers from 1 to $N$. These people already ate a lot of meat: the $k$-th person ate $A[k]$ kilograms of meat so far. Bajs will distribute his ham to the people in the ratio $B[1] : B[2] : \cdots : B[N]$, exactly in that order, but he doesn’t know the total amount (number of kilos) of ham which he will be distributing yet.

At the end of the slaughter, the Man of the Year will be declared. A ranking list is made according to the total kilos of meat eaten. Bajs impacts directly on this list by choosing the amount of ham to distribute. Although Bajs has been offered bribes many times, he refused each time, saying that he was an honest man who would not hurt a fly.

Bajs cares about order, because he’s a nice gentleman, and wants to have the order of people in the exact form of $1, 2, 3, \ldots , N$, respectively from the one who ate the most kilos of meat to those who ate less, allowing ties between participants. Help Bajs select the total amount of ham that he will distribute (in the ratio mentioned before) to achieve his intention.

-----Input-----
The first line of input contains an integer $N$ ($2 \geq N \geq 1000$), the number of competitors for the Man of the Year award. Each of the following $N$ lines contains integers $A[k]$ and $B[k]$, as mentioned in the text ($0 \geq A[k], B[k] \geq 10^6$). At least one of the numbers $B[k]$ will not be equal to 0.

-----Output-----
The first and only line of output must contain $-1$ if the required order cannot be achieved. Otherwise, output the required amount of ham in kilos, a real number between 0 and $10^7$ (inclusive). The answer should be correct up to either a relative or absolute error of $10^{-12}$. If there are multiple possible solutions, output any.

-----Examples-----
Sample Input 1:
3
7 1
3 2
10 0
Sample Output 1:
10.5

Sample Input 2:
3
2 1
4 0
0 3
Sample Output 2:
-1
user: Polly, the programming parrot, has escaped. As Johan is in desperate need for some programming skill, he wants to find her. The only clue that Johan has about the escape route is that Polly flew into his kitchen.

Johan has $N$ boxes on his kitchen table, and each box has a certain probability to contain the parrot. Each box also requires a certain amount of energy in order to be opened. Checking all the boxes would for sure find the disobedient parrot, but there is a problem. Johan is a very lazy person, and he wants to spend as little energy as possible while trying to find Polly. Aware of his own laziness, Johan decided that he will be satisfied with finding the parrot with at least probability $P$, as long as he minimizes the energy wasted in opening boxes. Can you help him?

-----Input-----
Input starts with a line with an integer $1 \leq N \leq 10^3$, the number of boxes in Johan’s kitchen and a floating point number $0 \leq P \leq 1$, the minimum probability with which Johan wants to find Polly. Then $N$ lines follow: each line containing an integer $0 \leq e_ i \leq 10^3$ describing the amount of energy needed to open box $i$, and a floating point number $0 \leq p_ i \leq 1$ describing the probability that Polly is inside box number $i$. All probabilities are given with at most four digits after the decimal point.

-----Output-----
Output should contain a single integer on a single line: the minimum amount of energy Johan has to waste while finding Polly with at least probability $P$.

-----Examples-----
Sample Input 1:
2 0.5
2 0.5
1 0.5
Sample Output 1:
1

Sample Input 2:
2 0.5
2 0.51
1 0.49
Sample Output 2:
2
user: A number of regional Internet Service Providers (ISPs), both big and small have recently been forced into a merger by the government, in an effort to improve service for all. Of course this has been decided without consulting you, the chief network infrastructure officer, and a deadline for when the ISPs should be merged have already been set.

You have a set of $n$ servers, each with a limited number of network sockets that can be used for connecting physically to other servers. Some servers are already linked up in the existing network architecture, and if server 0 is linked to server 2, then 2 is also linked to server 0 (as you use full duplex ethernet Cat6 cables). No server is directly connected to itself, and no two servers are directly linked with more than one connection.

You want to connect the servers to form a single network, such that all servers can reach each other through some sequence of connections. To make the set deadline, you have estimated that you only have time to make $k$ edits to the existing network infrastructure. An edit is either to remove an existing connection between two servers, or to add a new connection between two servers.

Can you connect all the servers to the same network using at most $k$ edits, within the limitations on the number of network sockets in each server?

-----Input-----
The first line of the input is three space separated integers $n$ ($1 \leq n \leq 100000$), the number of servers, $m$ ($0 \leq m \leq 200000$), the number of existing connections and $k$ ($0 \leq k \leq 50000$), the number of edits you have time to make. Then follows a line with $n$ integers $c_0, c_1, \ldots , c_ i, \ldots , c_{n-1}$, with the $i$’th number giving the number of network sockets for the $i$’th server (for all $i$ the capacity is bounded by $1 \leq c_ i < n$). Then $m$ lines follow, with two integers $u_ j$ and $v_ j$ each, giving the id’s of two servers that are connected in the old network architecture. Servers are $0$-indexed, i.e. for every $j$, it holds that $0 \leq u_ j, v_ j < n$. A server will never be connected to more servers than it has connection sockets.

-----Output-----
Output “yes” on a single line if the servers can be connected to one network by making $k$ or less edits, and “no” if it is not possible.

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

Sample Input 2:
5 4 4
1 1 2 2 2
0 1
2 3
3 4
4 2
Sample Output 2:
yes
user: Instead of paying attention in class, Luka is killing time playing with numbers again. This time, he wrote $N$ non-negative integers so that they form a ring (circle). After that he formed a new ring by adding to each number its two neighbours.

The teacher noticed this and took away the first piece of paper, with the original ring. This did not trouble Luka much because he knows he can use the other ring to reconstruct the original.

Write a program that solves Luka’s problem.

-----Input-----
The first line contains the integer $N$ $(3 \le N \le 10000)$, the number of integers in the ring.

Each of the following lines contains an integer less than $10^9$ (one billion). These numbers, in order, form the second ring.

The input will be such that a solution, although not necesarily unique, will always exist.

-----Output-----
Output the original ring on $N$ lines. The numbers must be non-negative.

Rotating the ring is not allowed. For example, the sum of the first three output numbers must be equal to the second number in the input ring.

Note: the solution need not be unique.

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

Sample Input 2:
4
20
15
17
14
Sample Output 2:
5
8
2
7
user: You might have heard about the friendly interactive shell - fish. It’s terminal based, but a bit friendlier to new users who can’t remember the commands they typed before. Fish can use your history of commands for command completion: pressing up (“^”) finds the last command you ran that shares the prefix you have already typed (a string is a prefix of itself).

If you press up again you go another step back in time, finding the second last command that shares the prefix of what you typed. If there are no more matching commands you stay at the oldest matching command.

When a series of consecutive up-presses is ended, either by typing some other character or by pressing enter to run the current command, the matching command from the history replaces everything that was typed in the current input.

When you press enter the current input is added to the command history. The command history does not contain up-characters (“^”) since command completion is always performed before the enter-key is processed.

The input to this problem contains the characters a user typed in a fish terminal, separated into lines. After each line, the user pressed enter. Your task is to output the resulting command history.

-----Input-----
The first line of the input contains a single number $n$, where $1 \leq n \leq 100000$.

Then follows $n$ lines with the user input. Each line consists of characters from the alphabet $\big [$a-zA-Z0-9.^-$\big ]$, i.e. English lower and upper case letters, digits, “-”, “ ”, “.” and the symbol “^” denoting the up key. No command begins or ends with a space.

The input is guaranteed to contain at most $10^6$ characters.

-----Output-----
For each command in the input output a line with the completed command (where all “^” are expanded). The total amount of output is guaranteed to be at most $10^6$ characters.

-----Sample Explanation-----
In Sample Input $1$ your first command is “python”. Your next command gets auto completed to “python”, and then you write “ main.py”, which is appended to your command making “python main.py”. The last command completes the first up press to “python main.py”, then “ -n 10” is typed which becomes “python main.py -n 10”.

In Sample Input $2$ your first two commands are “python” and “java”. The third command becomes auto completed to java, your previous command. The fourth command completes to “python” since it was your third last command. For your fifth command the third last command auto completed is now “java”.

In Sample Input $3$ no commands are in your history, so your up presses have no effect and only the non up presses count to your command making “python”.

-----Examples-----
Sample Input 1:
3
python
p^ main.py
^ -n 10
Sample Output 1:
python
python main.py
python main.py -n 10

Sample Input 2:
5
python
java
^
^^^
^^^
Sample Output 2:
python
java
java
python
java
user: It is a well-known fact that if you mix up the letters of a word, while leaving the first and last letters in their places, words still remain readable. For example, the sentence “tihs snetncee mkaes prfecet sesne”, makes perfect sense to most people. If you remove all spaces from a sentence, it still remains perfectly readable, see for example: “thissentencemakesperfectsense”, however if you combine these two things, first shuffling, then removing spaces, things get hard. The following sentence is harder to decipher: “tihssnetnceemkaesprfecetsesne”. You are given a sentence in the last form, together with a dictionary of valid words and are asked to decipher the text.

-----Input-----
 - One line with a string $s$: the sentence to decipher. The sentence consists of lowercase letters and has a length of at least $1$ and at most $1000$ characters.
 - One line with an integer $n$ with $1 \le n \le 10000$: the number of words in the dictionary.
 - $n$ lines with one word each. A word consists of lowercase letters and has a length of at least $1$ and at most $100$ characters. All the words are unique.

-----Output-----
Output one line with the deciphered sentence, if it is possible to uniquely decipher it. Otherwise “impossible” or “ambiguous”, depending on which is the case.

-----Examples-----
Sample Input 1:
tihssnetnceemkaesprfecetsesne
5
makes
perfect
sense
sentence
this
Sample Output 1:
this sentence makes perfect sense

Sample Input 2:
hitehre
2
there
hello
Sample Output 2:
impossible