meetforever/GRPO_CL huggingface dataset:
user: You are an evil sorcerer at a round table with $N$ sorcerers (including yourself). You can cast $M$ spells which have distinct powers $p_1, p_2, \ldots, p_M$.
You may perform the following operation any number of times (possibly zero):
- Assign a living sorcerer to each positive integer cyclically to your left starting from yourself ― the closest living sorcerer to your left is assigned to $1$, the next living sorcerer to the left is assigned to $2$ and so on. Note that each living sorcerer (including yourself) is assigned to an infinite number of integers.
- Choose a spell $j$ (possibly a spell you have chosen before) and kill the living sorcerer assigned to $p_j$. You may not cast a spell to kill yourself. 

What is the maximum number of sorcerers you can kill using zero or more operations?

-----Input-----
- The first line of the input contains a single integer $T$ denoting the number of test cases. The description of $T$ test cases follows.
- The first line of each test case contains two space-separated integers $N$ and $M$.
- The second line contains $M$ space-separated integers $p_1, p_2, \ldots, p_M$.

-----Output-----
For each test case, print a single line containing one integer ― the maximum number of sorcerers you can kill.

-----Constraints-----
- $1 \le T \le 1,000$
- $1 \le N \le 10^9$
- $1 \le M \le 3 \cdot 10^5$
- $1 \le p_i \le 10^9$ for each valid $i$
- $p_1, p_2, \ldots, p_N$ are pairwise distinct
- the sum of $M$ over all test cases does not exceed $3 \cdot 10^5$

-----Example Input-----
5
4 1
5
6 2
2 4
1 4
7 16 8 29
1000000000 1
998244353
1 1
20201220

-----Example Output-----
3
4
0
1755647
0

-----Explanation-----
Example case 1: The initial state is shown in the figure from the statement. We can first use spell $1$ and kill the $5$-th sorcerer to our left, i.e. sorcerer $2$. Now there are $3$ living sorcerers and the state is as shown in the following figure:

We can use spell $1$ again and kill the current $5$-th living sorcerer to our left, i.e. sorcerer $4$. Now there are $2$ living sorcerers and the state is:

Finally, we can use spell $1$ again and kill the only other living sorcerer, i.e. sorcerer $3$. Now, none of the other sorcerers are alive. As we cannot cast a spell to kill ourselves, we cannot improve the answer any further.
Example case 2: We can perform $4$ operations using the spell $p_1 = 2$ each time. We can also instead use $p_2 = 4$ in the first two operations and $p_1 = 2$ in the last two operations. Note that there may be multiple valid sequences of operations that lead to the best answer.
Example case 3: We cannot perform any operations using any of the given spells, so we are unable to kill any sorcerers.
Example case 4: We can perform $1,755,647$ operations, each of them using the spell $p_1 = 998,244,353$.
user: Teacher Sungjae wanted to hold a programming competition for his students where every participant need to be included into  team. The participants submitted their team names before the deadline. After the competition ran for half an hour, (It is assured that each registered team will submit absolutely once within half an hour) Sungjae mistakenly pressed a button that changed the order of the registered team names. Now in the submission list, order of the characters in the team's name doesn't matter. That means $abc$, $acb$, $bac$, $bca$, $cab$, $cba$ refers to the same team. The competition ran for two hours and then ended. Sungjae now counting each of the team's score and wants to print the registered team names and score. The scoreboard should be ordered based on scores in decreasing order and if two teams have same score, Sangjae would follow lexicographical order.
$N$.$B$. frequency of each character's in a registered team's name will not match with another team. 
That means two teams named $xoxo$ and $oxox$ is not possible. Because both of them have the same frequency of each of the characters (two 'o' and two 'x'). Similarly $abb$ and $bab$ is not possible (because both of them have one 'a' and two 'b').
It is ensured that only possible test cases will be given.

-----Input:-----Input:
- 
First line will contain $T$, number of testcases. Then the testcases follow. 
- 
The first line of each test case contains two integers , $N$ and $R$ - total number of submissions and the number of submissions within first half an hour. 
- 
Then $R$  lines follow: the i'th line contains a string $ti$, registered names of the teams and an integer $pi$, points they got on that submission.
- 
Then $N-R$  lines follow: the i-th line contains a string $ti$- the i-th team's name (in any order) in lowercase letter only and $pi$ -points they got on that submission.

-----Output:-----Output:
For each testcase,print the scoreboard.
That means print the teams name and their point according to their score in decreasing order and if some of them have same score,print the teams name in lexicographical order

-----Constraints-----Constraints
- $1 \leq T \leq 10$
- $1 \leq R \leq N \leq 1000$
- $1 \leq ti \leq 1000$
- $1 \leq pi \leq 10^6$
Sum of points ($pi$) of a team will not cross $10^9$.

-----Sample Input:-----Sample Input:
1
10 5
amigoes  1
bannermen 1
monarchy 4
outliers 5
iniciador 10
aegimos 2
iiiacdnor 1
eilorstu 1
gimosae 3
mnachroy 7

-----Sample Output:-----Sample Output:
iniciador 11
monarchy 11
amigoes 6
outliers 6
bannermen 1

-----Explanation:-----Explanation:

$It$ $is$ $assured$ $that$ $each$ $team$ $will$ $submit$ $once$ $within$ $first$ $half$ $an$ $hour$.That means -                                          

that kind of submissions isn't possible within first half an hour.
Dataset can be huge. Use faster I/O method.
user: On a planet called RUIZ LAND, which is ruled by the queen, Erika Ruiz. Each person on that planet has a strength value (strength value >0).
That planet has a special rule made by the queen that a boy and a girl will form a couple if their Hate value is a prime number where $Hate$ is given by the formula:-
Hate = (boy's strength value)  XOR (girl's strength value )
You are given $N$ numbers denoting the strength value of $N$ girls, and each of the $N$ girls has to form a couple with a boy such that sum of $Hate$ value of all the $N$ couples will be minimum.
You need to print the strength value of each boy, Where the boy at index $i$ will form a couple with the girl at index $i$, where $1 \leq i \leq N$.
Assume that you can always find at least one boy having that strength for each girl.

-----Input:-----
- First line will contain $N$, the number of Girls. 
- Next line contains $N$ numbers separated by space denoting strength value for each girl.

-----Output:-----
Print the required $N$ numbers denoting strength of boys.

-----Constraints-----
- $1 \leq N \leq 100000$
- $1 \leq A_i \leq 10^9$ , (where $1 \leq i \leq N$) and $A_i$ denotes strength of i'th girl.

-----Sample Input:-----
2
10 16

-----Sample Output:-----
8 18
user: Vasya has ordered a pizza delivery. The pizza can be considered a perfect circle. There were $n$ premade cuts in the pizza when it was delivered. Each cut is a straight segment connecting the center of the pizza with its boundary.
Let $O$ be the center of the pizza, $P_i$ be the endpoint of the $i$-th cut lying on the boundary, and $R$ be the point of the boundary straight to the right of $O$. Then the counterclockwise-measured angle $\angle ROP_i$ is equal to $a_i$ degrees, where $a_i$ is an integer between $0$ and $359$. Note that angles between $0$ and $180$ angles correspond to $P_i$ in the top half of the pizza, while angles between $180$ and $360$ angles correspond to the bottom half.
Vasya may cut his pizza a few more times, and the new cuts still have to be straight segments starting at the center. He wants to make the pizza separated into several equal slices, with each slice being a circular sector with no cuts inside of it. How many new cuts Vasya will have to make?

-----Input:-----
The first line of input contains $T$ , i.e number of test cases per file.
The first line of each test case contains a single integer $n-$  the numbers of premade cuts ($2 \leq n \leq 360$).
The second lines contains $n$ integers $a_1, \ldots, a_n-$ angles of the cuts $1, \ldots, n$ respectively ($0 \leq a_1 < \ldots, a_{n - 1} < 360$).

-----Output:-----
Print a single integer$-$ the smallest number of additional cuts Vasya has to make so that the pizza is divided into several equal slices.

-----Constraints-----
- $1  \leq T  \leq 36$
- $2 \leq n \leq 360$
- $0 \leq a_1 < \ldots, a_{n - 1} < 360$

-----Sample Input:-----
3
4 
0 90 180 270
2
90 210
2
0 1

-----Sample Output:-----
0
1
358

-----EXPLANATION:-----
In the first sample the pizza is already cut into four equal slices.
In the second sample the pizza will be cut into three equal slices after making one extra cut at $330$ degrees.
In the third sample Vasya will have to cut his pizza into $360$ pieces of $1$ degree angle each.
user: Chef has an array A consisting of N integers (1-based indexing). He asks you to perform the following operation M times:

	for i = 2 to N:
		Ai = Ai + Ai-1

Your task is to find the xth element of the array (i.e., Ax) after performing the above operation M times. As the answer could be large, please output it modulo 109 + 7.

-----Input-----
- The first line of input contains an integer T denoting the number of test cases.
- The first line of each test case contains three space-separated integers — N, x, and M — denoting the size of the array, index of the element you need to find, and the amount of times you need to repeat operation before finding the element, respectively. The second line contains N space-separated integers A1, A2, …, AN.

-----Output-----
For each test case, output a single line containing one integer: Ax modulo 109 + 7. 

-----Constraints-----
- 1 ≤ T ≤ 10
- 1 ≤ x ≤ N ≤ 105
- 1 ≤ M ≤ 1018 
- 1 ≤ Ai  ≤ 1018

-----Subtasks-----Subtask 1 (8 points):
- 1 ≤ x ≤ min{2, N}Subtask 2 (24 points):
- 1 ≤ N * M ≤ 106Subtask 3 (68 points):  No additional constraints

-----Example-----
Input:
2
3 2 3
1 2 3
3 3 3 
1 2 3

Output:
5
15

-----Explanation-----
Values in the array A:
- Before the operations: [1, 2, 3]
- After the first operation: [1, 3, 6]
- After the second operation: [1, 4, 10]
- After the third operation: [1, 5, 15]

Since input file can be fairly large (about 8 MB), it's recommended to use fast I/O (for example, in C++, use scanf/printf instead of cin/cout).
user: Ganesh lives in Gopalmath. He is looking for Jojo. So he decides to collect Aadhar Card Information of all the citizens of India from UIDAI.
Someone told Ganesh that the sum of all the digits of Jojo’s Aadhar number is divisible by 10 and it is greater than zero. After finding all Aadhar numbers which are divisible by 10, Jojo’s Aadhar number is $N$th smallest Aadhar number. Hence, Ganesh wants to find Jojo’s  Aadhar number which satisfies all of the above conditions. (In this chaotic world, Aadhar numbers can be any natural number.)
However, Guruji refused Ganesh to carry out this task, because he is weak in Maths. Therefore, Ganesh assigns this task to Paritoshbhai who possesses excellent Mathematical skills. Since Paritoshbhai is busy in his jewellery business, help him in this task.

-----Input:-----
- The first line of the input contains a single integer T denoting the number of test cases. The description of T test cases follows.
- The first and only line of each test case contains a single integer N.

-----Output:-----
For each test case, print a single line containing Aadhar number of Jojo.

-----Constraints-----
- $1 \leq T \leq 1000$
- $1 \leq N \leq 10^{100,000}$

-----Sample Input:-----
1
3

-----Sample Output:-----
37
user: Teddy and Tracy like to play a game based on strings. The game is as follows. Initially, Tracy writes a long random string on a whiteboard. Then, each player starting with Teddy makes turn alternately. Each turn, the player must erase a contiguous substring that exists in the dictionary. The dictionary consists of N words.
Of course, the player that can't erase any substring in his turn loses the game, and the other player is declared the winner.
Note that after a substring R is erased, the remaining substring becomes separated, i.e. they cannot erase a word that occurs partially to the left of R and partially to the right of R.

Determine the winner of the game, assuming that both players play optimally.

-----Input-----
The first line contains a single integer T, the number of test cases. T test cases follow. The first line of each testcase contains a string S, the string Tracy writes on the whiteboard. The next line contains a single integer N. N lines follow. The i-th line contains a single string wi, the i-th word in the dictionary.

-----Output-----
For each test case, output a single line containing the name of the winner of the game.

-----Example-----
Input:
3
codechef
2
code
chef
foo
1
bar
mississippi
4
ssissi
mippi
mi
ppi

Output:
Tracy
Tracy
Teddy

-----Constraints-----
- 1 <= T <= 5
- 1 <= N <= 30
- 1 <= |S| <= 30
- 1 <= |wi| <= 30
- S and wi contain only characters 'a'-'z'
user: Mathison recently inherited an ancient papyrus that contained some text. Unfortunately, the text was not a
pangram. Now, Mathison has a particular liking for holoalphabetic strings and the text bothers him. The good news is that Mathison can buy letters from the local store in order to turn his text into a pangram.

However, each letter has a price and Mathison is not very rich. Can you help Mathison find the cheapest way to obtain a pangram?

-----Input-----
The first line of the input file will contain one integer, T, representing the number of tests.
Each test will be formed from two lines. The first one contains 26 space-separated integers, representing the prices of all letters.
The second will contain Mathison's initial text (a string of N lowercase letters).

-----Output-----
The output file will contain T lines, one for each test. Each line will contain the answer for the corresponding test.

-----Constraints and notes-----
- 1 ≤ T ≤ 10
- 1 ≤ N ≤ 50,000
- All prices are natural numbers between 1 and 1,000,000 (i.e. 106).
- A pangram is a string that contains every letter of the Latin alphabet at least once.
- All purchased letters are added to the end of the string.

-----Subtaks-----
Subtask #1 (30 points):

- N = 1

Subtask #2 (70 points):

- Original constraints

-----Example-----
Input:
2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
abcdefghijklmopqrstuvwz
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
thequickbrownfoxjumpsoverthelazydog

Output:
63
0

-----Explanation-----
First test
There are three letters missing from the original string: n (price 14), x (price 24), and y (price 25).
Therefore the answer is 14 + 24 + 25 = 63.

Second test
No letter is missing so there is no point in buying something. The answer is 0.
user: You are given a sequence of integers $A_1,A_2,…,A_N$ and a magical non-zero integer $x$
You have to select a subsegment of sequence A (possibly empty), and replace the elements in that subsegment after dividing them by x.
Formally, replace any one subsegment $A_l, A_{l+1}, ..., A_r$ with $A_l/x, A_{l+1}/x, ..., A_r/x$ where $l \leq r$
What is the minimum possible sum you can obtain?
Note: The given operation can only be performed once

-----Input -----
- The first line of the input contains two positive integer n denoting the size of array, and x
denoting the magical integer
- Next line contains $N$ space separated integers

-----Output-----
Single line containing one real number, denoting the minimum possible sum you can obtain. Your answer will be considered correct if its absolute or relative error does not exceed $10^{-2}$

-----Constraints-----
- $1 \leq n \leq 10^3$
- $1 \leq |x| \leq 10^3$
- $ |A_i| \leq 10^3$

-----Sample Input-----
3 2
1 -2 3

-----Sample Output-----
0.5

-----Explanation-----
Array 1 -2 3, selecting subsegment {3}, you get 1 -2 1.5, which gives $sum=0.5$
user: You are given a string $S$. Find the number of ways to choose an unordered pair of non-overlapping non-empty substrings of this string (let's denote them by $s_1$ and $s_2$ in such a way that $s_2$ starts after $s_1$ ends) such that their concatenation $s_1 + s_2$ is a palindrome.
Two pairs $(s_1, s_2)$ and $(s_1', s_2')$ are different if $s_1$ is chosen at a different position from $s_1'$ or $s_2$ is chosen at a different position from $s_2'$.

-----Input-----
The first and only line of the input contains a single string $S$.

-----Output-----
Print a single line containing one integer — the number of ways to choose a valid pair of substrings.

-----Constraints-----
- $1 \le |S| \le 1,000$
- $S$ contains only lowercase English letters

-----Subtasks-----
Subtask #1 (25 points): $|S| \le 100$
Subtask #2 (75 points): original constraints

-----Example Input-----
abba

-----Example Output-----
7

-----Explanation-----
The following pairs of substrings can be chosen: ("a", "a"), ("a", "ba"), ("a", "bba"), ("ab", "a"), ("ab", "ba"), ("abb", "a"), ("b", "b").
user: Chef is playing a game on the non-negative x-axis. It takes him $1$ second to reach from $i^{th}$ position to $(i-1)^{th}$ position or $(i+1)^{th}$ position. The chef never goes to the negative x-axis. Also, Chef doesn't stop at any moment of time.
The movement of chef can be described as follows.
- At the start he is standing at $x=0$ at time $0$.
- In the first round, he moves towards $x=1$ and comes back to the $x=0$ position. 
- In the second round, he moves towards the $x=2$ and comes back again to $x=0$. 
- Generalizing, in the $k^{th}$ round, he moves from $x=0$ to $x=k$ and then returns back to $x=0$ at the end of the round. This goes on as the game progresses.
For Example, the path of Chef for $3^{rd}$ round is given below.
$0 - 1 - 2 - 3 - 2 - 1 - 0$ 
The overall path followed by Chef would look somewhat like this:
$0 - 1 - 0 - 1 - 2 - 1 - 0 - 1 - 2 - 3 - 2 - 1 - 0 - 1 - 2 - 3 - 4 - 3 - …$
You are given two non-negative integers $N$ and $K$. You have to tell the time at which Chef arrives at $x=N$ for the $K^{th}$ time. 
Note - Chef can not skip a position while visiting the positions.

-----Input:-----
- The first line contains $T$ the number of test cases. Then the test cases follow.
- Each test case contains a single line of two integers $N$ and $K$.

-----Output:-----
For each test case, print a single line containing one integer -- the time taken by the chef to arrive at $x=N$ for the $K^{th}$ time by modulo $1,000,000,007$.

-----Constraints-----
- $1 \le T \le 10^5$
- $0 \le N \le 10^9$
- $1 \le K \le 10^9$

-----Sample Input:-----
5
0 1
1 1
2 1
1 3
4 6

-----Sample Output:-----
0
1
4
5
46

-----Explanation:-----
Test Case 1:

Chef starts the journey from the $N = 0$ at time $t = 0$ and it's the first time $(K = 1)$, he is here. So, the answer is $0$.  
Test Case 2:

Chef starts the journey from the $N = 0$ at time $t = 0$ then goes to $N = 1$ at $t = 1$ and it's the first time $(K = 1)$, he is here. So, the answer is $1$.
Test Case 4:

The path followed by Chef to reach $1$ for the third time is given below.
$0 - 1 - 0 - 1 - 2 - 1$

He reaches $1$ for the third time at $t=5$.
user: Chef Ada is building a new restaurant in the following way:
- First, $N$ points $X_1, X_2, \ldots, X_N$ are chosen on the $x$-axis.
- Then, $N$ columns (numbered $1$ through $N$) are made. For simplicity, the columns are represented as vertical segments; for each valid $i$, the height of the $i$-th segment is $H_i$.
- Ada assigns a column to each of the points $X_1, X_2, \ldots, X_N$ in an arbitrary way (each column must be assigned to exactly one point).
- Finally, Ada constructs the roof of the restaurant, represented by a polyline with $N$ vertices. Let's denote the column assigned to the $i$-th point by $P_i$. For each valid $i$, the $i$-th of these vertices is $(X_i, H_{P_i})$, i.e. the polyline joins the tops of the columns from left to right.
Ada wants the biggest restaurant. Help her choose the positions of the columns in such a way that the area below the roof is the biggest possible. Formally, she wants to maximise the area of the polygon whose perimeter is formed by the roof and the segments $(X_N, H_{P_N}) - (X_N, 0) - (X_1, 0) - (X_1, H_{P_1})$. Let $S$ be this maximum area; you should compute $2 \cdot S$ (it is guaranteed that $2 \cdot S$ is an integer).

-----Input-----
- The first line of the input contains a single integer $T$ denoting the number of test cases. The description of $T$ test cases follows.
- The first line of each test case contains a single integer $N$.
- $N$ lines follow. For each valid $i$, the $i$-th of these lines contains two space-separated integers $X_i$ and $H_i$.

-----Output-----
For each test case, print a single line containing one integer $2 \cdot S$.

-----Constraints-----
- $1 \le T \le 3 \cdot 10^5$
- $2 \le N \le 10^5$
- $0 \le X_1 < X_2 < \ldots < X_N \le 2 \cdot 10^9$
- $1 \le H_i \le 10^9$ for each valid $i$
- the sum of $N$ over all test cases does not exceed $10^6$

-----Example Input-----
1
5
1 1
2 2
3 3
4 4
5 5

-----Example Output-----
27

-----Explanation-----
user: Roman has no idea, why this problem is called Stone. He also has no idea on how to solve the followong problem: given array of N integers A and a number K. During a turn the maximal value over all Ai is chosen, let's call it MAX. Then Ai = 
MAX - Ai is done for every 1 <= i <= N. Help Roman to find out how will the array look like after K turns.

-----Input-----
The numbers N and K are given in the first line of an input. Then N integers are given in the second line which denote the array A. 

-----Output-----
Output N numbers on a single line. It should be the array A after K turns.

-----Constraints-----

- 1 <= N <= 105
- 0 <= K <= 109
- Ai does not exceed 2 * 109 by it's absolute value.

-----Example-----
Input:
4 1
5 -1 7 0

Output:
2 8 0 7
user: Chef just got a box of chocolates as his birthday gift. The box contains $N$ chocolates in a row (numbered $1$ through $N$), where $N$ is even. For each valid $i$, the $i$-th chocolate has a sweetness value $W_i$.
Chef wants to eat all the chocolates in the first half of the box and leave all chocolates in the second half uneaten. Since he does not like chocolates that are too sweet, he will be unhappy if at least one of the chocolates he eats has the maximum sweetness among all the chocolates in the box.
A right cyclic shift by $k$ chocolates ($0 \le k < N$) consists of moving the last $k$ chocolates in the row to the beginning in the same order and moving each of the remaining $N-k$ chocolates $k$ places to the right. Before eating the first half of the chocolates, Chef wants to perform some right cyclic shift in such a way that he will not be unhappy after eating them. Find the number of ways to do this, i.e. the number of valid integers $k$ such that if Chef performs the right cyclic shift by $k$ chocolates and then eats the first half of the chocolates in the box, he does not become unhappy.

-----Input-----
- The first line of the input contains a single integer $T$ denoting the number of test cases. The description of $T$ test cases follows.
- The first line of each test case contains a single integer $N$.
- The second line contains $N$ space-separated integers $W_1, W_2, \ldots, W_N$.

-----Output-----
For each test case, print a single line containing one integer ― the number of shifts for which Chef does not become unhappy.

-----Constraints-----
- $1 \le T \le 5$
- $1 \le N \le 10^5$
- $N$ is even
- $1 \le W_i \le 10^5$ for each valid $i$

-----Example Input-----
2
6
1 1 2 1 1 1
6
1 1 2 1 1 2

-----Example Output-----
3
0

-----Explanation-----
Example case 1: The three valid right shifts and the contents of the box for these shifts are:
- shift by $k = 1$: $(1, 1, 1, 2, 1, 1)$
- shift by $k = 2$: $(1, 1, 1, 1, 2, 1)$
- shift by $k = 3$: $(1, 1, 1, 1, 1, 2)$