meetforever/GRPO_CL huggingface dataset:
user: Let x be a string of length at least 1.
We will call x a good string, if for any string y and any integer k (k \geq 2), the string obtained by concatenating k copies of y is different from x.
For example, a, bbc and cdcdc are good strings, while aa, bbbb and cdcdcd are not.
Let w be a string of length at least 1.
For a sequence F=(\,f_1,\,f_2,\,...,\,f_m) consisting of m elements,
we will call F a good representation of w, if the following conditions are both satisfied:
 - For any i \, (1 \leq i \leq m), f_i is a good string.
 - The string obtained by concatenating f_1,\,f_2,\,...,\,f_m in this order, is w.
For example, when w=aabb, there are five good representations of w:
 - (aabb)
 - (a,abb)
 - (aab,b)
 - (a,ab,b)
 - (a,a,b,b)
Among the good representations of w, the ones with the smallest number of elements are called the best representations of w.
For example, there are only one best representation of w=aabb: (aabb).
You are given a string w. Find the following:
 - the number of elements of a best representation of w
 - the number of the best representations of w, modulo 1000000007 \, (=10^9+7)
(It is guaranteed that a good representation of w always exists.)

-----Constraints-----
 - 1 \leq |w| \leq 500000 \, (=5 \times 10^5)
 - w consists of lowercase letters (a-z).

-----Partial Score-----
 - 400 points will be awarded for passing the test set satisfying 1 \leq |w| \leq 4000.

-----Input-----
The input is given from Standard Input in the following format:
w

-----Output-----
Print 2 lines.
 - In the first line, print the number of elements of a best representation of w.
 - In the second line, print the number of the best representations of w, modulo 1000000007.

-----Sample Input-----
aab

-----Sample Output-----
1
1
user: There is an integer sequence of length 2^N: A_0, A_1, ..., A_{2^N-1}. (Note that the sequence is 0-indexed.)
For every integer K satisfying 1 \leq K \leq 2^N-1, solve the following problem:
 - Let i and j be integers. Find the maximum value of A_i + A_j where 0 \leq i < j \leq 2^N-1 and (i or j) \leq K.
Here, or denotes the bitwise OR.

-----Constraints-----
 - 1 \leq N \leq 18
 - 1 \leq A_i \leq 10^9
 - All values in input are integers.

-----Input-----
Input is given from Standard Input in the following format:
N
A_0 A_1 ... A_{2^N-1}

-----Output-----
Print 2^N-1 lines.
In the i-th line, print the answer of the problem above for K=i.

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

-----Sample Output-----
3
4
5

For K=1, the only possible pair of i and j is (i,j)=(0,1), so the answer is A_0+A_1=1+2=3.
For K=2, the possible pairs of i and j are (i,j)=(0,1),(0,2).
When (i,j)=(0,2), A_i+A_j=1+3=4. This is the maximum value, so the answer is 4.
For K=3, the possible pairs of i and j are (i,j)=(0,1),(0,2),(0,3),(1,2),(1,3),(2,3) .
When (i,j)=(1,2), A_i+A_j=2+3=5. This is the maximum value, so the answer is 5.
user: There are N bags, each containing two white balls. The i-th box contains two balls with integers x_i and y_i written on them, respectively.
For each of these bags, you will paint one of the balls red, and paint the other blue.
Afterwards, the 2N balls will be classified according to color.
Then, we will define the following:
 - R_{max}: the maximum integer written on a ball painted in red
 - R_{min}: the minimum integer written on a ball painted in red
 - B_{max}: the maximum integer written on a ball painted in blue
 - B_{min}: the minimum integer written on a ball painted in blue
Find the minimum possible value of (R_{max} - R_{min}) \times (B_{max} - B_{min}).

-----Constraints-----
 - 1 ≤ N ≤ 200,000
 - 1 ≤ x_i, y_i ≤ 10^9

-----Input-----
Input is given from Standard Input in the following format:
N
x_1 y_1
x_2 y_2
:
x_N y_N

-----Output-----
Print the minimum possible value.

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

-----Sample Output-----
15

The optimal solution is to paint the balls with x_1, x_2, y_3 red, and paint the balls with y_1, y_2, x_3 blue.
user: You are given sequences A and B consisting of non-negative integers.
The lengths of both A and B are N, and the sums of the elements in A and B are equal.
The i-th element in A is A_i, and the i-th element in B is B_i.
Tozan and Gezan repeats the following sequence of operations:
 - If A and B are equal sequences, terminate the process.
 - Otherwise, first Tozan chooses a positive element in A and decrease it by 1.
 - Then, Gezan chooses a positive element in B and decrease it by 1.
 - Then, give one candy to Takahashi, their pet.
Tozan wants the number of candies given to Takahashi until the process is terminated to be as large as possible, while Gezan wants it to be as small as possible.
Find the number of candies given to Takahashi when both of them perform the operations optimally.

-----Constraints-----
 - 1 \leq N \leq 2 × 10^5
 - 0 \leq A_i,B_i \leq 10^9(1\leq i\leq N)
 - The sums of the elements in A and B are equal.
 - All values in input are integers.

-----Input-----
Input is given from Standard Input in the following format:
N
A_1 B_1
:
A_N B_N

-----Output-----
Print the number of candies given to Takahashi when both Tozan and Gezan perform the operations optimally.

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

-----Sample Output-----
2

When both Tozan and Gezan perform the operations optimally, the process will proceed as follows:
 - Tozan decreases A_1 by 1.
 - Gezan decreases B_1 by 1.
 - One candy is given to Takahashi.
 - Tozan decreases A_2 by 1.
 - Gezan decreases B_1 by 1.
 - One candy is given to Takahashi.
 - As A and B are equal, the process is terminated.
user: You are given a string S consisting of lowercase English letters.
Determine whether we can turn S into a palindrome by repeating the operation of swapping two adjacent characters. If it is possible, find the minimum required number of operations.

-----Constraints-----
 - 1 \leq |S| \leq 2 × 10^5
 - S consists of lowercase English letters.

-----Input-----
Input is given from Standard Input in the following format:
S

-----Output-----
If we cannot turn S into a palindrome, print -1. Otherwise, print the minimum required number of operations.

-----Sample Input-----
eel

-----Sample Output-----
1

We can turn S into a palindrome by the following operation:
 - Swap the 2-nd and 3-rd characters. S is now ele.
user: There is a simple undirected graph with N vertices and M edges.
The vertices are numbered 1 through N, and the edges are numbered 1 through M.
Edge i connects Vertex U_i and V_i.
Also, Vertex i has two predetermined integers A_i and B_i.
You will play the following game on this graph.
First, choose one vertex and stand on it, with W yen (the currency of Japan) in your pocket.
Here, A_s \leq W must hold, where s is the vertex you choose.
Then, perform the following two kinds of operations any number of times in any order:
 - Choose one vertex v that is directly connected by an edge to the vertex you are standing on, and move to vertex v. Here, you need to have at least A_v yen in your pocket when you perform this move.
 - Donate B_v yen to the vertex v you are standing on. Here, the amount of money in your pocket must not become less than 0 yen.
You win the game when you donate once to every vertex.
Find the smallest initial amount of money W that enables you to win the game.

-----Constraints-----
 - 1 \leq N \leq 10^5
 - N-1 \leq M \leq 10^5
 - 1 \leq A_i,B_i \leq 10^9
 - 1 \leq U_i < V_i \leq N
 - The given graph is connected and simple (there is at most one edge between any pair of vertices).

-----Input-----
Input is given from Standard Input in the following format:
N M
A_1 B_1
A_2 B_2
:
A_N B_N
U_1 V_1
U_2 V_2
:
U_M V_M

-----Output-----
Print the smallest initial amount of money W that enables you to win the game.

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

-----Sample Output-----
6

If you have 6 yen initially, you can win the game as follows:
 - Stand on Vertex 4. This is possible since you have not less than 6 yen.
 - Donate 2 yen to Vertex 4. Now you have 4 yen.
 - Move to Vertex 3. This is possible since you have not less than 4 yen.
 - Donate 1 yen to Vertex 3. Now you have 3 yen.
 - Move to Vertex 2. This is possible since you have not less than 1 yen.
 - Move to Vertex 1. This is possible since you have not less than 3 yen.
 - Donate 1 yen to Vertex 1. Now you have 2 yen.
 - Move to Vertex 2. This is possible since you have not less than 1 yen.
 - Donate 2 yen to Vertex 2. Now you have 0 yen.
If you have less than 6 yen initially, you cannot win the game. Thus, the answer is 6.
user: There are N towns located in a line, conveniently numbered 1 through N. Takahashi the merchant is going on a travel from town 1 to town N, buying and selling apples.
Takahashi will begin the travel at town 1, with no apple in his possession. The actions that can be performed during the travel are as follows:
 - Move: When at town i (i < N), move to town i + 1.
 - Merchandise: Buy or sell an arbitrary number of apples at the current town. Here, it is assumed that one apple can always be bought and sold for A_i yen (the currency of Japan) at town i (1 ≦ i ≦ N), where A_i are distinct integers. Also, you can assume that he has an infinite supply of money.
For some reason, there is a constraint on merchandising apple during the travel: the sum of the number of apples bought and the number of apples sold during the whole travel, must be at most T. (Note that a single apple can be counted in both.)
During the travel, Takahashi will perform actions so that the profit of the travel is maximized. Here, the profit of the travel is the amount of money that is gained by selling apples, minus the amount of money that is spent on buying apples. Note that we are not interested in apples in his possession at the end of the travel.
Aoki, a business rival of Takahashi, wants to trouble Takahashi by manipulating the market price of apples. Prior to the beginning of Takahashi's travel, Aoki can change A_i into another arbitrary non-negative integer A_i' for any town i, any number of times. The cost of performing this operation is |A_i - A_i'|. After performing this operation, different towns may have equal values of A_i.
Aoki's objective is to decrease Takahashi's expected profit by at least 1 yen. Find the minimum total cost to achieve it. You may assume that Takahashi's expected profit is initially at least 1 yen.

-----Constraints-----
 - 1 ≦ N ≦ 10^5
 - 1 ≦ A_i ≦ 10^9 (1 ≦ i ≦ N)
 - A_i are distinct.
 - 2 ≦ T ≦ 10^9
 - In the initial state, Takahashi's expected profit is at least 1 yen.

-----Input-----
The input is given from Standard Input in the following format:
N T
A_1 A_2 ... A_N

-----Output-----
Print the minimum total cost to decrease Takahashi's expected profit by at least 1 yen.

-----Sample Input-----
3 2
100 50 200

-----Sample Output-----
1

In the initial state, Takahashi can achieve the maximum profit of 150 yen as follows:
 - Move from town 1 to town 2.
 - Buy one apple for 50 yen at town 2.
 - Move from town 2 to town 3.
 - Sell one apple for 200 yen at town 3.
If, for example, Aoki changes the price of an apple at town 2 from 50 yen to 51 yen, Takahashi will not be able to achieve the profit of 150 yen. The cost of performing this operation is 1, thus the answer is 1.
There are other ways to decrease Takahashi's expected profit, such as changing the price of an apple at town 3 from 200 yen to 199 yen.
user: Let N be a positive even number.
We have a permutation of (1, 2, ..., N), p = (p_1, p_2, ..., p_N).
Snuke is constructing another permutation of (1, 2, ..., N), q, following the procedure below.
First, let q be an empty sequence.
Then, perform the following operation until p becomes empty:
 - Select two adjacent elements in p, and call them x and y in order. Remove x and y from p (reducing the length of p by 2), and insert x and y, preserving the original order, at the beginning of q.
When p becomes empty, q will be a permutation of (1, 2, ..., N).
Find the lexicographically smallest permutation that can be obtained as q.

-----Constraints-----
 - N is an even number.
 - 2 ≤ N ≤ 2 × 10^5
 - p is a permutation of (1, 2, ..., N).

-----Input-----
Input is given from Standard Input in the following format:
N
p_1 p_2 ... p_N

-----Output-----
Print the lexicographically smallest permutation, with spaces in between.

-----Sample Input-----
4
3 2 4 1

-----Sample Output-----
3 1 2 4

The solution above is obtained as follows:pq(3, 2, 4, 1)()↓↓(3, 1)(2, 4)↓↓()(3, 1, 2, 4)
user: Snuke has a rooted tree with N+1 vertices.
The vertices are numbered 0 through N, and Vertex 0 is the root of the tree.
The parent of Vertex i (1 \leq i \leq N) is Vertex p_i.
Besides this tree, Snuke also has an box which is initially empty and many marbles, and playing with them.
The play begins with placing one marble on some of the vertices, then proceeds as follows:
 - If there is a marble on Vertex 0, move the marble into the box.
 - Move each marble from the vertex to its parent (all at once).
 - For each vertex occupied by two or more marbles, remove all the marbles from the vertex.
 - If there exists a vertex with some marbles, go to Step 1. Otherwise, end the play.
There are 2^{N+1} ways to place marbles on some of the vertices.
For each of them, find the number of marbles that will be in the box at the end of the play, and compute the sum of all those numbers modulo 1,000,000,007.

-----Constraints-----
 - 1 \leq N < 2 \times 10^{5}
 - 0 \leq p_i < i

-----Partial Scores-----
 - In the test set worth 400 points, N < 2{,}000.

-----Input-----
Input is given from Standard Input in the following format:
N
p_1 p_2 ... p_{N}

-----Output-----
Print the answer.

-----Sample Input-----
2
0 0

-----Sample Output-----
8

When we place a marble on both Vertex 1 and 2, there will be multiple marbles on Vertex 0 by step 2. In such a case, these marbles will be removed instead of being moved to the box.
user: You are in charge of controlling a dam. The dam can store at most L liters of water. Initially, the dam is empty. Some amount of water flows into the dam every morning, and any amount of water may be discharged every night, but this amount needs to be set so that no water overflows the dam the next morning.
It is known that v_i liters of water at t_i degrees Celsius will flow into the dam on the morning of the i-th day.
You are wondering about the maximum possible temperature of water in the dam at noon of each day, under the condition that there needs to be exactly L liters of water in the dam at that time. For each i, find the maximum possible temperature of water in the dam at noon of the i-th day. Here, consider each maximization separately, that is, the amount of water discharged for the maximization of the temperature on the i-th day, may be different from the amount of water discharged for the maximization of the temperature on the j-th day (j≠i).
Also, assume that the temperature of water is not affected by anything but new water that flows into the dam. That is, when V_1 liters of water at T_1 degrees Celsius and V_2 liters of water at T_2 degrees Celsius are mixed together, they will become V_1+V_2 liters of water at \frac{T_1*V_1+T_2*V_2}{V_1+V_2} degrees Celsius, and the volume and temperature of water are not affected by any other factors.

-----Constraints-----
 - 1≤ N ≤ 5*10^5
 - 1≤ L ≤ 10^9
 - 0≤ t_i ≤ 10^9(1≤i≤N)
 - 1≤ v_i ≤ L(1≤i≤N)
 - v_1 = L
 - L, each t_i and v_i are integers.

-----Input-----
Input is given from Standard Input in the following format:
N L
t_1 v_1
t_2 v_2
:
t_N v_N

-----Output-----
Print N lines. The i-th line should contain the maximum temperature such that it is possible to store L liters of water at that temperature in the dam at noon of the i-th day.
Each of these values is accepted if the absolute or relative error is at most 10^{-6}.

-----Sample Input-----
3 10
10 10
20 5
4 3

-----Sample Output-----
10.0000000
15.0000000
13.2000000

 - On the first day, the temperature of water in the dam is always 10 degrees: the temperature of the only water that flows into the dam on the first day.
 - 10 liters of water at 15 degrees of Celsius can be stored on the second day, by discharging 5 liters of water on the night of the first day, and mix the remaining water with the water that flows into the dam on the second day.
 - 10 liters of water at 13.2 degrees of Celsius can be stored on the third day, by discharging 8 liters of water on the night of the first day, and mix the remaining water with the water that flows into the dam on the second and third days.
user: Alice lives on a line. Today, she will travel to some place in a mysterious vehicle.
Initially, the distance between Alice and her destination is D. When she input a number x to the vehicle, it will travel in the direction of the destination by a distance of x if this move would shorten the distance between the vehicle and the destination, and it will stay at its position otherwise. Note that the vehicle may go past the destination when the distance between the vehicle and the destination is less than x.
Alice made a list of N numbers. The i-th number in this list is d_i. She will insert these numbers to the vehicle one by one.
However, a mischievous witch appeared. She is thinking of rewriting one number in the list so that Alice will not reach the destination after N moves.
She has Q plans to do this, as follows:
 - Rewrite only the q_i-th number in the list with some integer so that Alice will not reach the destination.
Write a program to determine whether each plan is feasible.

-----Constraints-----
 - 1≤ N ≤ 5*10^5
 - 1≤ Q ≤ 5*10^5
 - 1≤ D ≤ 10^9
 - 1≤ d_i ≤ 10^9(1≤i≤N)
 - 1≤ q_i ≤ N(1≤i≤Q)
 - D and each d_i are integers.

-----Input-----
Input is given from Standard Input in the following format:
N D
d_1 d_2 ... d_N
Q
q_1 q_2 ... q_Q

-----Output-----
Print Q lines. The i-th line should contain YES if the i-th plan is feasible, and NO otherwise.

-----Sample Input-----
4 10
3 4 3 3
2
4 3

-----Sample Output-----
NO
YES

For the first plan, Alice will already arrive at the destination by the first three moves, and therefore the answer is NO.
For the second plan, rewriting the third number in the list with 5 will prevent Alice from reaching the destination as shown in the following figure, and thus the answer is YES.
user: Snuke's town has a subway system, consisting of N stations and M railway lines. The stations are numbered 1 through N. Each line is operated by a company. Each company has an identification number.
The i-th ( 1 \leq i \leq M ) line connects station p_i and q_i bidirectionally. There is no intermediate station. This line is operated by company c_i.
You can change trains at a station where multiple lines are available.
The fare system used in this subway system is a bit strange. When a passenger only uses lines that are operated by the same company, the fare is 1 yen (the currency of Japan). Whenever a passenger changes to a line that is operated by a different company from the current line, the passenger is charged an additional fare of 1 yen. In a case where a passenger who changed from some company A's line to another company's line changes to company A's line again, the additional fare is incurred again.
Snuke is now at station 1 and wants to travel to station N by subway. Find the minimum required fare.

-----Constraints-----
 - 2 \leq N \leq 10^5
 - 0 \leq M \leq 2×10^5
 - 1 \leq p_i \leq N (1 \leq i \leq M)
 - 1 \leq q_i \leq N (1 \leq i \leq M)
 - 1 \leq c_i \leq 10^6 (1 \leq i \leq M)
 - p_i \neq q_i (1 \leq i \leq M)

-----Input-----
The input is given from Standard Input in the following format:
N M
p_1 q_1 c_1
:
p_M q_M c_M

-----Output-----
Print the minimum required fare. If it is impossible to get to station N by subway, print -1 instead.

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

-----Sample Output-----
1

Use company 1's lines: 1 → 2 → 3. The fare is 1 yen.
user: Sig has built his own keyboard. Designed for ultimate simplicity, this keyboard only has 3 keys on it: the 0 key, the 1 key and the backspace key.
To begin with, he is using a plain text editor with this keyboard. This editor always displays one string (possibly empty). Just after the editor is launched, this string is empty. When each key on the keyboard is pressed, the following changes occur to the string:
 - The 0 key: a letter 0 will be inserted to the right of the string.
 - The 1 key: a letter 1 will be inserted to the right of the string.
 - The backspace key: if the string is empty, nothing happens. Otherwise, the rightmost letter of the string is deleted.
Sig has launched the editor, and pressed these keys N times in total. As a result, the editor displays a string s. Find the number of such ways to press the keys, modulo 10^9 + 7.

-----Constraints-----
 - 1 ≦ N ≦ 5000
 - 1 ≦ |s| ≦ N
 - s consists of the letters 0 and 1.

-----Partial Score-----
 - 400 points will be awarded for passing the test set satisfying 1 ≦ N ≦ 300.

-----Input-----
The input is given from Standard Input in the following format:
N
s

-----Output-----
Print the number of the ways to press the keys N times in total such that the editor displays the string s in the end, modulo 10^9+7.

-----Sample Input-----
3
0

-----Sample Output-----
5

We will denote the backspace key by B. The following 5 ways to press the keys will cause the editor to display the string 0 in the end: 00B, 01B, 0B0, 1B0, BB0. In the last way, nothing will happen when the backspace key is pressed.
user: Takahashi has an ability to generate a tree using a permutation (p_1,p_2,...,p_n) of (1,2,...,n), in the following process:
First, prepare Vertex 1, Vertex 2, ..., Vertex N.
For each i=1,2,...,n, perform the following operation:
 - If p_i = 1, do nothing.
 - If p_i \neq 1, let j' be the largest j such that p_j < p_i. Span an edge between Vertex i and Vertex j'.
Takahashi is trying to make his favorite tree with this ability.
His favorite tree has n vertices from Vertex 1 through Vertex n, and its i-th edge connects Vertex v_i and w_i.
Determine if he can make a tree isomorphic to his favorite tree by using a proper permutation.
If he can do so, find the lexicographically smallest such permutation.

-----Notes-----
For the definition of isomorphism of trees, see wikipedia. Intuitively, two trees are isomorphic when they are the "same" if we disregard the indices of their vertices.

-----Constraints-----
 - 2 \leq n \leq 10^5
 - 1 \leq v_i, w_i \leq n
 - The given graph is a tree.

-----Input-----
Input is given from Standard Input in the following format:
n
v_1 w_1
v_2 w_2
:
v_{n-1} w_{n-1}

-----Output-----
If there is no permutation that can generate a tree isomorphic to Takahashi's favorite tree, print -1.
If it exists, print the lexicographically smallest such permutation, with spaces in between.

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

-----Sample Output-----
1 2 4 5 3 6

If the permutation (1, 2, 4, 5, 3, 6) is used to generate a tree, it looks as follows:

This is isomorphic to the given graph.