meetforever/GRPO_CL huggingface dataset:
user: Fennec and Snuke are playing a board game.
On the board, there are N cells numbered 1 through N, and N-1 roads, each connecting two cells. Cell a_i is adjacent to Cell b_i through the i-th road. Every cell can be reached from every other cell by repeatedly traveling to an adjacent cell. In terms of graph theory, the graph formed by the cells and the roads is a tree.
Initially, Cell 1 is painted black, and Cell N is painted white. The other cells are not yet colored.
Fennec (who goes first) and Snuke (who goes second) alternately paint an uncolored cell.
More specifically, each player performs the following action in her/his turn:
 - Fennec: selects an uncolored cell that is adjacent to a black cell, and paints it black.
 - Snuke: selects an uncolored cell that is adjacent to a white cell, and paints it white.
A player loses when she/he cannot paint a cell. Determine the winner of the game when Fennec and Snuke play optimally.

-----Constraints-----
 - 2 \leq N \leq 10^5
 - 1 \leq a_i, b_i \leq N
 - The given graph is a tree.

-----Input-----
Input is given from Standard Input in the following format:
N
a_1 b_1
:
a_{N-1} b_{N-1}

-----Output-----
If Fennec wins, print Fennec; if Snuke wins, print Snuke.

-----Sample Input-----
7
3 6
1 2
3 1
7 4
5 7
1 4

-----Sample Output-----
Fennec

For example, if Fennec first paints Cell 2 black, she will win regardless of Snuke's moves.
user: Let us consider the following operations on a string consisting of A and B:
 - Select a character in a string. If it is A, replace it with BB. If it is B, replace with AA.
 - Select a substring that is equal to either AAA or BBB, and delete it from the string.
For example, if the first operation is performed on ABA and the first character is selected, the string becomes BBBA.
If the second operation is performed on BBBAAAA and the fourth through sixth characters are selected, the string becomes BBBA.
These operations can be performed any number of times, in any order.
You are given two string S and T, and q queries a_i, b_i, c_i, d_i.
For each query, determine whether S_{a_i} S_{{a_i}+1} ... S_{b_i}, a substring of S, can be made into T_{c_i} T_{{c_i}+1} ... T_{d_i}, a substring of T.

-----Constraints-----
 - 1 \leq |S|, |T| \leq 10^5
 - S and T consist of letters A and B.
 - 1 \leq q \leq 10^5
 - 1 \leq a_i \leq b_i \leq |S|
 - 1 \leq c_i \leq d_i \leq |T|

-----Input-----
Input is given from Standard Input in the following format:
S
T
q
a_1 b_1 c_1 d_1
...
a_q b_q c_q d_q

-----Output-----
Print q lines. The i-th line should contain the response to the i-th query. If S_{a_i} S_{{a_i}+1} ... S_{b_i} can be made into T_{c_i} T_{{c_i}+1} ... T_{d_i}, print YES. Otherwise, print NO.

-----Sample Input-----
BBBAAAABA
BBBBA
4
7 9 2 5
7 9 1 4
1 7 2 5
1 7 2 4

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

The first query asks whether the string ABA can be made into BBBA.
As explained in the problem statement, it can be done by the first operation.
The second query asks whether ABA can be made into BBBB, and the fourth query asks whether BBBAAAA can be made into BBB.
Neither is possible.
The third query asks whether the string BBBAAAA can be made into BBBA.
As explained in the problem statement, it can be done by the second operation.
user: Snuke loves constructing integer sequences.
There are N piles of stones, numbered 1 through N.
The pile numbered i consists of a_i stones.
Snuke will construct an integer sequence s of length Σa_i, as follows:
 - Among the piles with the largest number of stones remaining, let x be the index of the pile with the smallest index. Append x to the end of s.
 - Select a pile with one or more stones remaining, and remove a stone from that pile.
 - If there is a pile with one or more stones remaining, go back to step 1. Otherwise, terminate the process.
We are interested in the lexicographically smallest sequence that can be constructed. For each of the integers 1,2,3,...,N, how many times does it occur in the lexicographically smallest sequence?

-----Constraints-----
 - 1 ≤ N ≤ 10^{5}
 - 1 ≤ a_i ≤ 10^{9}

-----Input-----
The input is given from Standard Input in the following format:
N
a_1 a_2 ... a_{N}

-----Output-----
Print N lines. The i-th line should contain the number of the occurrences of the integer i in the lexicographically smallest sequence that can be constructed.

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

-----Sample Output-----
2
1

The lexicographically smallest sequence is constructed as follows:
 - Since the pile with the largest number of stones remaining is pile 2, append 2 to the end of s. Then, remove a stone from pile 2.
 - Since the piles with the largest number of stones remaining are pile 1 and 2, append 1 to the end of s (we take the smallest index). Then, remove a stone from pile 2.
 - Since the pile with the largest number of stones remaining is pile 1, append 1 to the end of s. Then, remove a stone from pile 1.
The resulting sequence is (2,1,1). In this sequence, 1 occurs twice, and 2 occurs once.
user: Snuke has decided to play a game, where the player runs a railway company.
There are M+1 stations on Snuke Line, numbered 0 through M.
A train on Snuke Line stops at station 0 and every d-th station thereafter, where d is a predetermined constant for each train.
For example, if d = 3, the train stops at station 0, 3, 6, 9, and so forth.
There are N kinds of souvenirs sold in areas around Snuke Line. The i-th kind of souvenirs can be purchased when the train stops at one of the following stations: stations l_i, l_i+1, l_i+2, ..., r_i.
There are M values of d, the interval between two stops, for trains on Snuke Line: 1, 2, 3, ..., M.
For each of these M values, find the number of the kinds of souvenirs that can be purchased if one takes a train with that value of d at station 0.
Here, assume that it is not allowed to change trains.

-----Constraints-----
 - 1 ≦ N ≦ 3 × 10^{5}
 - 1 ≦ M ≦ 10^{5}
 - 1 ≦ l_i ≦ r_i ≦ M

-----Input-----
The input is given from Standard Input in the following format:
N M
l_1 r_1
:
l_{N} r_{N}

-----Output-----
Print the answer in M lines. The i-th line should contain the maximum number of the kinds of souvenirs that can be purchased if one takes a train stopping every i-th station.

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

-----Sample Output-----
3
2
2

 - If one takes a train stopping every station, three kinds of souvenirs can be purchased: kind 1, 2 and 3.
 - If one takes a train stopping every second station, two kinds of souvenirs can be purchased: kind 1 and 2.
 - If one takes a train stopping every third station, two kinds of souvenirs can be purchased: kind 2 and 3.
user: When Misha hits his favorite gym, he comes across an interesting problem with the barbell. In the gym, someone always leaves the weight plates in the strangest places you can imagine and sometime it's difficult to equip the barbell the way you want. Let's imagine that you have N weight plates placed in any order (remember that any gym has no more than K different types of weight plates and all weights are square-free). As a preliminary step towards solving this problem, Misha wants to simulate a simple gym, and for this purpose you have to deal with some queries:

- [1 I X] Set the weight of the ith weight plate to value X.

- [2 L R] Reverse the sequence of weight plates in the interval from L to R, where 1 ≤ L ≤ R ≤ N.

- [3 L R W] Check the interval from L to R to find out if you can make the weight W using only weight plates on this interval. (Note: this type of query will appear no more than P times)

Please help Misha in solving this problem. 

-----Input-----
First line of input contains the number of weight plates N, and number of queries Q. Next line contains N integers w1, w2, ..., wN, where wi is the weight of the ith weight plate. Next Q lines contain some queries described above. 

-----Output-----
For all queries of the third type: print "Yes" if your check returns a positive outcome, and "No" otherwise.

-----Constraints-----
- 1 ≤ N, W, Q ≤ 105
- K ≤ 10
- P ≤ 1000
- All numbers in the input are positive integers and ≤ 105.
- All the weights are square-free.

-----Subtasks-----
- Subtask 1: 1 ≤ N ≤ 103, 1 ≤ W ≤ 103, Q = 1 - 10 pts.

- Subtask 2: 1 ≤ N ≤ 103, 1 ≤ W ≤ 103, 1 ≤ Q ≤ 103, P ≤ 100 - 15 pts
- Subtask 3: 1 ≤ N ≤ 104, 1 ≤ W ≤ 104, 1 ≤ Q ≤ 104, P ≤ 300 - 25 pts.

- Subtask 4: 1 ≤ N ≤ 105, 1 ≤ W ≤ 105, 1 ≤ Q ≤ 105, K ≤ 2 - 20 pts.

- Subtask 5: Original constraints - 30 pts.

-----Example-----First
Input:5 10
1 2 3 5 6
3 2 3 3
3 2 3 4
3 2 3 5
2 2 5
3 2 4 8
1 2 1
3 2 4 8
2 1 4 
3 2 4 3 
3 1 5 7 

Output:Yes
No
Yes
Yes
Yes
No
YesSecond
Input:3 4
2013 2015 2017
3 1 3 4030
1 1 111
3 1 3 4030
3 1 2 111

Output:Yes
No
Yes

-----Explanation:-----First test explanation (step by step)
1 2 3 5 6
3 2 3 3 ([2, 3] 3=3 => Yes)
3 2 3 4 ([2, 3] can't make 4 => No)
3 2 3 5 ([2, 3] 2+3=5 => Yes)
2 2 5 (Reverse: [1, 6, 5, 3, 2])
3 2 4 8 ([6, 5, 3] 5+3=8 => Yes)
1 2 1 (Set: [1, 1, 5, 3, 2])
3 2 4 8 ([1, 5, 3] 5+3=8 => Yes)
2 1 4  (Reverse: [3, 5, 1, 1, 2])
3 2 4 3 ([5, 1, 1] can't make 3 => No)
3 1 5 7 ([3, 5, 1, 1, 2] 2+1+1+3=7 => Yes)
user: You are given an array A, consisting of N integers and an array B, consisting of M integers.
The subsequence of A is the array that can be obtained by picking the elements at the arbitrary sorted set of positions from A.
Your task is to count the number of such subsequences C of A that:

- C contains exactly M elements.
- The array (C+B) is non-decreasing. Here by + operation, we mean element-wise sum. 

For example, the array (4, 8, 5) plus the array (10, 20, 30) is (14, 28, 35).

Formally, (C+B) is an array of size M such that (C+B)i = Ci + Bi.

In case some subsequence appears more that once, you should counts it as many times as it appears.

Formally, two subarrays of an array a, (ai_1, ai_2, ... ,ai_n) and (aj_1, aj_2, ... ,aj_m) will be considered different if either their lengths are different i.e. n != m or there exists an index k such that such that i_k  != j_k.

Since the answer can be very large, we ask you to calculate it, modulo 109+7.

-----Input-----
The first line of input contains a pair of space separated integers N and M, denoting the number of elements in the array A and the number of elements in the array B.
The second line contains N space-separated integers Ai, denoting the array A.
The third line contains M space-separated integers Bj, denoting the array B.

-----Output-----
Output a single line containing the number of subsequences C as asked in the problem, modulo 109+7.

-----Constraints-----
- 1 ≤ Ai, Bi ≤ 109
- 1 ≤ M ≤ N

-----Subtasks-----
- Subtask #1 (33 points): 1 ≤ N ≤ 50, 1 ≤ M ≤ 5
- Subtask #2 (33 points): 1 ≤ N ≤ 500, 1 ≤ M ≤ 50
- Subtask #3 (34 points): 1 ≤ N ≤ 2000, 1 ≤ M ≤ 1000

-----Example-----
Input #1:
5 3
1 5 2 4 7
7 9 6

Output #1:
4

Input #2:
4 2
7 7 7 7
3 4

Output #2:
6

-----Explanation-----
Example case 1. The suitable subsequences are (1, 2, 7), (1, 4, 7), (5, 4, 7), (2, 4, 7).
Example case 2. The suitable subsequence is (7, 7), and it appears 6 times:

- at indices (1, 2)
- at indices (1, 3)
- at indices (1, 4)
- at indices (2, 3)
- at indices (2, 4)
- at indices (3, 4)

So, the answer is 6.
user: You are given a string s of length n.
Does a tree with n vertices that satisfies the following conditions exist?
 - The vertices are numbered 1,2,..., n.
 - The edges are numbered 1,2,..., n-1, and Edge i connects Vertex u_i and v_i.
 - If the i-th character in s is 1, we can have a connected component of size i by removing one edge from the tree.
 - If the i-th character in s is 0, we cannot have a connected component of size i by removing any one edge from the tree.
If such a tree exists, construct one such tree.

-----Constraints-----
 - 2 \leq n \leq 10^5
 - s is a string of length n consisting of 0 and 1.

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

-----Output-----
If a tree with n vertices that satisfies the conditions does not exist, print -1.
If a tree with n vertices that satisfies the conditions exist, print n-1 lines.
The i-th line should contain u_i and v_i with a space in between.
If there are multiple trees that satisfy the conditions, any such tree will be accepted.

-----Sample Input-----
1111

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

It is impossible to have a connected component of size n after removing one edge from a tree with n vertices.
user: There is an undirected tree of $n$ vertices, connected by $n-1$ bidirectional edges. There is also a snake stuck inside of this tree. Its head is at vertex $a$ and its tail is at vertex $b$. The snake's body occupies all vertices on the unique simple path between $a$ and $b$.

The snake wants to know if it can reverse itself  — that is, to move its head to where its tail started, and its tail to where its head started. Unfortunately, the snake's movements are restricted to the tree's structure.

In an operation, the snake can move its head to an adjacent vertex not currently occupied by the snake. When it does this, the tail moves one vertex closer to the head, so that the length of the snake remains unchanged. Similarly, the snake can also move its tail to an adjacent vertex not currently occupied by the snake. When it does this, the head moves one unit closer to the tail. [Image] Let's denote a snake position by $(h,t)$, where $h$ is the index of the vertex with the snake's head, $t$ is the index of the vertex with the snake's tail. This snake can reverse itself with the movements $(4,7)\to (5,1)\to (4,2)\to (1, 3)\to (7,2)\to (8,1)\to (7,4)$. 

Determine if it is possible to reverse the snake with some sequence of operations.


-----Input-----

The first line contains a single integer $t$ ($1\le t\le 100$)  — the number of test cases. The next lines contain descriptions of test cases.

The first line of each test case contains three integers $n,a,b$ ($2\le n\le 10^5,1\le a,b\le n,a\ne b$).

Each of the next $n-1$ lines contains two integers $u_i,v_i$ ($1\le u_i,v_i\le n,u_i\ne v_i$), indicating an edge between vertices $u_i$ and $v_i$. It is guaranteed that the given edges form a tree.

It is guaranteed that the sum of $n$ across all test cases does not exceed $10^5$.


-----Output-----

For each test case, output "YES" if it is possible for the snake to reverse itself, or "NO" otherwise.


-----Example-----
Input
4
8 4 7
1 2
2 3
1 4
4 5
4 6
1 7
7 8
4 3 2
4 3
1 2
2 3
9 3 5
1 2
2 3
3 4
1 5
5 6
6 7
1 8
8 9
16 15 12
1 2
2 3
1 4
4 5
5 6
6 7
4 8
8 9
8 10
10 11
11 12
11 13
13 14
10 15
15 16

Output
YES
NO
NO
YES



-----Note-----

The first test case is pictured above.

In the second test case, the tree is a path. We can show that the snake cannot reverse itself.

In the third test case, we can show that the snake cannot reverse itself.

In the fourth test case, an example solution is:

$(15,12)\to (16,11)\to (15,13)\to (10,14)\to (8,13)\to (4,11)\to (1,10)$

$\to (2,8)\to (3,4)\to (2,5)\to (1,6)\to (4,7)\to (8,6)\to (10,5)$

$\to (11,4)\to (13,8)\to (14,10)\to (13,15)\to (11,16)\to (12,15)$.
user: There are $n$ cities in Berland and some pairs of them are connected by two-way roads. It is guaranteed that you can pass from any city to any other, moving along the roads. Cities are numerated from $1$ to $n$.

Two fairs are currently taking place in Berland — they are held in two different cities $a$ and $b$ ($1 \le a, b \le n$; $a \ne b$).

Find the number of pairs of cities $x$ and $y$ ($x \ne a, x \ne b, y \ne a, y \ne b$) such that if you go from $x$ to $y$ you will have to go through both fairs (the order of visits doesn't matter). Formally, you need to find the number of pairs of cities $x,y$ such that any path from $x$ to $y$ goes through $a$ and $b$ (in any order).

Print the required number of pairs. The order of two cities in a pair does not matter, that is, the pairs $(x,y)$ and $(y,x)$ must be taken into account only once.


-----Input-----

The first line of the input contains an integer $t$ ($1 \le t \le 4\cdot10^4$) — the number of test cases in the input. Next, $t$ test cases are specified.

The first line of each test case contains four integers $n$, $m$, $a$ and $b$ ($4 \le n \le 2\cdot10^5$, $n - 1 \le m \le 5\cdot10^5$, $1 \le a,b \le n$, $a \ne b$) — numbers of cities and roads in Berland and numbers of two cities where fairs are held, respectively.

The following $m$ lines contain descriptions of roads between cities. Each of road description contains a pair of integers $u_i, v_i$ ($1 \le u_i, v_i \le n$, $u_i \ne v_i$) — numbers of cities connected by the road.

Each road is bi-directional and connects two different cities. It is guaranteed that from any city you can pass to any other by roads. There can be more than one road between a pair of cities.

The sum of the values of $n$ for all sets of input data in the test does not exceed $2\cdot10^5$. The sum of the values of $m$ for all sets of input data in the test does not exceed $5\cdot10^5$.


-----Output-----

Print $t$ integers — the answers to the given test cases in the order they are written in the input.


-----Example-----
Input
3
7 7 3 5
1 2
2 3
3 4
4 5
5 6
6 7
7 5
4 5 2 3
1 2
2 3
3 4
4 1
4 2
4 3 2 1
1 2
2 3
4 1

Output
4
0
1
user: Consider the infinite sequence $s$ of positive integers, created by repeating the following steps:

  Find the lexicographically smallest triple of positive integers $(a, b, c)$ such that   $a \oplus b \oplus c = 0$, where $\oplus$ denotes the bitwise XOR operation.  $a$, $b$, $c$ are not in $s$.  Here triple of integers $(a_1, b_1, c_1)$ is considered to be lexicographically smaller than triple $(a_2, b_2, c_2)$ if sequence $[a_1, b_1, c_1]$ is lexicographically smaller than sequence $[a_2, b_2, c_2]$.  Append $a$, $b$, $c$ to $s$ in this order.  Go back to the first step. 

You have integer $n$. Find the $n$-th element of $s$.

You have to answer $t$ independent test cases.

A sequence $a$ is lexicographically smaller than a sequence $b$ if in the first position where $a$ and $b$ differ, the sequence $a$ has a smaller element than the corresponding element in $b$.


-----Input-----

The first line contains a single integer $t$ ($1 \le t \le 10^5$) — the number of test cases.

Each of the next $t$ lines contains a single integer $n$ ($1\le n \le 10^{16}$) — the position of the element you want to know.


-----Output-----

In each of the $t$ lines, output the answer to the corresponding test case.


-----Example-----
Input
9
1
2
3
4
5
6
7
8
9

Output
1
2
3
4
8
12
5
10
15



-----Note-----

The first elements of $s$ are $1, 2, 3, 4, 8, 12, 5, 10, 15, \dots $
user: Zookeeper is playing a game. In this game, Zookeeper must use bombs to bomb a string that consists of letters 'A' and 'B'. He can use bombs to bomb a substring which is either "AB" or "BB". When he bombs such a substring, the substring gets deleted from the string and the remaining parts of the string get concatenated.

For example, Zookeeper can use two such operations: AABABBA $\to$ AABBA $\to$ AAA.

Zookeeper wonders what the shortest string he can make is. Can you help him find the length of the shortest string?


-----Input-----

Each test contains multiple test cases. The first line contains a single integer $t$ $(1 \leq t \leq 20000)$  — the number of test cases. The description of the test cases follows.

Each of the next $t$ lines contains a single test case each, consisting of a non-empty string $s$: the string that Zookeeper needs to bomb. It is guaranteed that all symbols of $s$ are either 'A' or 'B'.

It is guaranteed that the sum of $|s|$ (length of $s$) among all test cases does not exceed $2 \cdot 10^5$.


-----Output-----

For each test case, print a single integer: the length of the shortest string that Zookeeper can make.


-----Example-----
Input
3
AAA
BABA
AABBBABBBB

Output
3
2
0



-----Note-----

For the first test case, you can't make any moves, so the answer is $3$.

For the second test case, one optimal sequence of moves is BABA $\to$ BA. So, the answer is $2$.

For the third test case, one optimal sequence of moves is AABBBABBBB $\to$ AABBBABB $\to$ AABBBB $\to$ ABBB $\to$ AB $\to$ (empty string). So, the answer is $0$.
user: Petr likes to come up with problems about randomly generated data. This time problem is about random permutation. He decided to generate a random permutation this way: he takes identity permutation of numbers from $1$ to $n$ and then $3n$ times takes a random pair of different elements and swaps them. Alex envies Petr and tries to imitate him in all kind of things. Alex has also come up with a problem about random permutation. He generates a random permutation just like Petr but swaps elements $7n+1$ times instead of $3n$ times. Because it is more random, OK?!

You somehow get a test from one of these problems and now you want to know from which one.


-----Input-----

In the first line of input there is one integer $n$ ($10^{3} \le n \le 10^{6}$).

In the second line there are $n$ distinct integers between $1$ and $n$ — the permutation of size $n$ from the test.

It is guaranteed that all tests except for sample are generated this way: First we choose $n$ — the size of the permutation. Then we randomly choose a method to generate a permutation — the one of Petr or the one of Alex. Then we generate a permutation using chosen method.


-----Output-----

If the test is generated via Petr's method print "Petr" (without quotes). If the test is generated via Alex's method print "Um_nik" (without quotes).


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

Output
Petr



-----Note-----

Please note that the sample is not a valid test (because of limitations for $n$) and is given only to illustrate input/output format. Your program still has to print correct answer to this test to get AC.

Due to randomness of input hacks in this problem are forbidden.
user: This is the hard version of the problem. The difference between the versions is the constraint on $n$ and the required number of operations. You can make hacks only if all versions of the problem are solved.

There are two binary strings $a$ and $b$ of length $n$ (a binary string is a string consisting of symbols $0$ and $1$). In an operation, you select a prefix of $a$, and simultaneously invert the bits in the prefix ($0$ changes to $1$ and $1$ changes to $0$) and reverse the order of the bits in the prefix.

For example, if $a=001011$ and you select the prefix of length $3$, it becomes $011011$. Then if you select the entire string, it becomes $001001$.

Your task is to transform the string $a$ into $b$ in at most $2n$ operations. It can be proved that it is always possible.


-----Input-----

The first line contains a single integer $t$ ($1\le t\le 1000$)  — the number of test cases. Next $3t$ lines contain descriptions of test cases.

The first line of each test case contains a single integer $n$ ($1\le n\le 10^5$)  — the length of the binary strings.

The next two lines contain two binary strings $a$ and $b$ of length $n$.

It is guaranteed that the sum of $n$ across all test cases does not exceed $10^5$.


-----Output-----

For each test case, output an integer $k$ ($0\le k\le 2n$), followed by $k$ integers $p_1,\ldots,p_k$ ($1\le p_i\le n$). Here $k$ is the number of operations you use and $p_i$ is the length of the prefix you flip in the $i$-th operation.


-----Example-----
Input
5
2
01
10
5
01011
11100
2
01
01
10
0110011011
1000110100
1
0
1

Output
3 1 2 1
6 5 2 5 3 1 2
0
9 4 1 2 10 4 1 2 1 5
1 1



-----Note-----

In the first test case, we have $01\to 11\to 00\to 10$.

In the second test case, we have $01011\to 00101\to 11101\to 01000\to 10100\to 00100\to 11100$.

In the third test case, the strings are already the same. Another solution is to flip the prefix of length $2$, which will leave $a$ unchanged.