meetforever/GRPO_CL huggingface dataset:
user: Ayush, Ashish and Vivek are busy preparing a new problem for the next Codeforces round and need help checking if their test cases are valid.

Each test case consists of an integer $n$ and two arrays $a$ and $b$, of size $n$. If after some (possibly zero) operations described below, array $a$ can be transformed into array $b$, the input is said to be valid. Otherwise, it is invalid.

An operation on array $a$ is:   select an integer $k$ $(1 \le k \le \lfloor\frac{n}{2}\rfloor)$  swap the prefix of length $k$ with the suffix of length $k$ 

For example, if array $a$ initially is $\{1, 2, 3, 4, 5, 6\}$, after performing an operation with $k = 2$, it is transformed into $\{5, 6, 3, 4, 1, 2\}$.

Given the set of test cases, help them determine if each one is valid or invalid.


-----Input-----

The first line contains one integer $t$ $(1 \le t \le 500)$ — the number of test cases. The description of each test case is as follows.

The first line of each test case contains a single integer $n$ $(1 \le n \le 500)$ — the size of the arrays.

The second line of each test case contains $n$ integers $a_1$, $a_2$, ..., $a_n$ $(1 \le a_i \le 10^9)$ — elements of array $a$.

The third line of each test case contains $n$ integers $b_1$, $b_2$, ..., $b_n$ $(1 \le b_i \le 10^9)$ — elements of array $b$.


-----Output-----

For each test case, print "Yes" if the given input is valid. Otherwise print "No".

You may print the answer in any case.


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

Output
yes
yes
No
yes
No



-----Note-----

For the first test case, we can swap prefix $a[1:1]$ with suffix $a[2:2]$ to get $a=[2, 1]$.

For the second test case, $a$ is already equal to $b$.

For the third test case, it is impossible since we cannot obtain $3$ in $a$.

For the fourth test case, we can first swap prefix $a[1:1]$ with suffix $a[4:4]$ to obtain $a=[2, 2, 3, 1]$. Now we can swap prefix $a[1:2]$ with suffix $a[3:4]$ to obtain $a=[3, 1, 2, 2]$.

For the fifth test case, it is impossible to convert $a$ to $b$.
user: Dora the explorer has decided to use her money after several years of juicy royalties to go shopping. What better place to shop than Nlogonia?

There are $n$ stores numbered from $1$ to $n$ in Nlogonia. The $i$-th of these stores offers a positive integer $a_i$.

Each day among the last $m$ days Dora bought a single integer from some of the stores. The same day, Swiper the fox bought a single integer from all the stores that Dora did not buy an integer from on that day.

Dora considers Swiper to be her rival, and she considers that she beat Swiper on day $i$ if and only if the least common multiple of the numbers she bought on day $i$ is strictly greater than the least common multiple of the numbers that Swiper bought on day $i$.

The least common multiple (LCM) of a collection of integers is the smallest positive integer that is divisible by all the integers in the collection.

However, Dora forgot the values of $a_i$. Help Dora find out if there are positive integer values of $a_i$ such that she beat Swiper on every day. You don't need to find what are the possible values of $a_i$ though.

Note that it is possible for some values of $a_i$ to coincide in a solution.


-----Input-----

The first line contains integers $m$ and $n$ ($1\leq m \leq 50$, $1\leq n \leq 10^4$) — the number of days and the number of stores.

After this $m$ lines follow, the $i$-th line starts with an integer $s_i$ ($1\leq s_i \leq n-1$), the number of integers Dora bought on day $i$, followed by $s_i$ distinct integers, the indices of the stores where Dora bought an integer on the $i$-th day. The indices are between $1$ and $n$.


-----Output-----

Output must consist of a single line containing "possible" if there exist positive integers $a_i$ such that for each day the least common multiple of the integers bought by Dora is strictly greater than the least common multiple of the integers bought by Swiper on that day. Otherwise, print "impossible".

Note that you don't have to restore the integers themselves.


-----Examples-----
Input
2 5
3 1 2 3
3 3 4 5

Output
possible

Input
10 10
1 1
1 2
1 3
1 4
1 5
1 6
1 7
1 8
1 9
1 10

Output
impossible



-----Note-----

In the first sample, a possible choice for the values of the $a_i$ is $3, 4, 3, 5, 2$. On the first day, Dora buys the integers $3, 4$ and $3$, whose LCM is $12$, while Swiper buys integers $5$ and $2$, whose LCM is $10$. On the second day, Dora buys $3, 5$ and $2$, whose LCM is $30$, and Swiper buys integers $3$ and $4$, whose LCM is $12$.
user: Little Artem has invented a time machine! He could go anywhere in time, but all his thoughts of course are with computer science. He wants to apply this time machine to a well-known data structure: multiset.

Artem wants to create a basic multiset of integers. He wants these structure to support operations of three types:

  Add integer to the multiset. Note that the difference between set and multiset is that multiset may store several instances of one integer.  Remove integer from the multiset. Only one instance of this integer is removed. Artem doesn't want to handle any exceptions, so he assumes that every time remove operation is called, that integer is presented in the multiset.  Count the number of instances of the given integer that are stored in the multiset. 

But what about time machine? Artem doesn't simply apply operations to the multiset one by one, he now travels to different moments of time and apply his operation there. Consider the following example.

  First Artem adds integer 5 to the multiset at the 1-st moment of time.  Then Artem adds integer 3 to the multiset at the moment 5.  Then Artem asks how many 5 are there in the multiset at moment 6. The answer is 1.  Then Artem returns back in time and asks how many integers 3 are there in the set at moment 4. Since 3 was added only at moment 5, the number of integers 3 at moment 4 equals to 0.  Then Artem goes back in time again and removes 5 from the multiset at moment 3.  Finally Artyom asks at moment 7 how many integers 5 are there in the set. The result is 0, since we have removed 5 at the moment 3. 

Note that Artem dislikes exceptions so much that he assures that after each change he makes all delete operations are applied only to element that is present in the multiset. The answer to the query of the third type is computed at the moment Artem makes the corresponding query and are not affected in any way by future changes he makes.

Help Artem implement time travellers multiset.


-----Input-----

The first line of the input contains a single integer n (1 ≤ n ≤ 100 000) — the number of Artem's queries.

Then follow n lines with queries descriptions. Each of them contains three integers a_{i}, t_{i} and x_{i} (1 ≤ a_{i} ≤ 3, 1 ≤ t_{i}, x_{i} ≤ 10^9) — type of the query, moment of time Artem travels to in order to execute this query and the value of the query itself, respectively. It's guaranteed that all moments of time are distinct and that after each operation is applied all operations of the first and second types are consistent.


-----Output-----

For each ask operation output the number of instances of integer being queried at the given moment of time.


-----Examples-----
Input
6
1 1 5
3 5 5
1 2 5
3 6 5
2 3 5
3 7 5

Output
1
2
1

Input
3
1 1 1
2 2 1
3 3 1

Output
0
user: You are given an array a consisting of n positive integers. You pick two integer numbers l and r from 1 to n, inclusive (numbers are picked randomly, equiprobably and independently). If l > r, then you swap values of l and r. You have to calculate the expected value of the number of unique elements in segment of the array from index l to index r, inclusive (1-indexed).


-----Input-----

The first line contains one integer number n (1 ≤ n ≤ 10^6). The second line contains n integer numbers a_1, a_2, ... a_{n} (1 ≤ a_{i} ≤ 10^6) — elements of the array.


-----Output-----

Print one number — the expected number of unique elements in chosen segment. 

Your answer will be considered correct if its absolute or relative error doesn't exceed 10^{ - 4} — formally, the answer is correct if $\operatorname{min}(|x - y|, \frac{|x - y|}{x}) \leq 10^{-4}$, where x is jury's answer, and y is your answer.


-----Examples-----
Input
2
1 2

Output
1.500000

Input
2
2 2

Output
1.000000
user: Little girl Margarita is a big fan of competitive programming. She especially loves problems about arrays and queries on them.

Recently, she was presented with an array $a$ of the size of $10^9$ elements that is filled as follows:   $a_1 = -1$  $a_2 = 2$  $a_3 = -3$  $a_4 = 4$  $a_5 = -5$  And so on ... 

That is, the value of the $i$-th element of the array $a$ is calculated using the formula $a_i = i \cdot (-1)^i$.

She immediately came up with $q$ queries on this array. Each query is described with two numbers: $l$ and $r$. The answer to a query is the sum of all the elements of the array at positions from $l$ to $r$ inclusive.

Margarita really wants to know the answer to each of the requests. She doesn't want to count all this manually, but unfortunately, she couldn't write the program that solves the problem either. She has turned to you — the best programmer.

Help her find the answers!


-----Input-----

The first line contains a single integer $q$ ($1 \le q \le 10^3$) — the number of the queries.

Each of the next $q$ lines contains two integers $l$ and $r$ ($1 \le l \le r \le 10^9$) — the descriptions of the queries.


-----Output-----

Print $q$ lines, each containing one number — the answer to the query. 


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

Output
-2
-2
-5
4
-1



-----Note-----

In the first query, you need to find the sum of the elements of the array from position $1$ to position $3$. The sum is equal to $a_1 + a_2 + a_3 = -1 + 2 -3 = -2$.

In the second query, you need to find the sum of the elements of the array from position $2$ to position $5$. The sum is equal to $a_2 + a_3 + a_4 + a_5 = 2 -3 + 4 - 5 = -2$.

In the third query, you need to find the sum of the elements of the array from position $5$ to position $5$. The sum is equal to $a_5 = -5$.

In the fourth query, you need to find the sum of the elements of the array from position $4$ to position $4$. The sum is equal to $a_4 = 4$.

In the fifth query, you need to find the sum of the elements of the array from position $2$ to position $3$. The sum is equal to $a_2 + a_3 = 2 - 3 = -1$.
user: You are given two integers $a$ and $b$, and $q$ queries. The $i$-th query consists of two numbers $l_i$ and $r_i$, and the answer to it is the number of integers $x$ such that $l_i \le x \le r_i$, and $((x \bmod a) \bmod b) \ne ((x \bmod b) \bmod a)$. Calculate the answer for each query.

Recall that $y \bmod z$ is the remainder of the division of $y$ by $z$. For example, $5 \bmod 3 = 2$, $7 \bmod 8 = 7$, $9 \bmod 4 = 1$, $9 \bmod 9 = 0$.


-----Input-----

The first line contains one integer $t$ ($1 \le t \le 100$) — the number of test cases. Then the test cases follow.

The first line of each test case contains three integers $a$, $b$ and $q$ ($1 \le a, b \le 200$; $1 \le q \le 500$).

Then $q$ lines follow, each containing two integers $l_i$ and $r_i$ ($1 \le l_i \le r_i \le 10^{18}$) for the corresponding query.


-----Output-----

For each test case, print $q$ integers — the answers to the queries of this test case in the order they appear.


-----Example-----
Input
2
4 6 5
1 1
1 3
1 5
1 7
1 9
7 10 2
7 8
100 200

Output
0 0 0 2 4 
0 91
user: During the lesson small girl Alyona works with one famous spreadsheet computer program and learns how to edit tables.

Now she has a table filled with integers. The table consists of n rows and m columns. By a_{i}, j we will denote the integer located at the i-th row and the j-th column. We say that the table is sorted in non-decreasing order in the column j if a_{i}, j ≤ a_{i} + 1, j for all i from 1 to n - 1.

Teacher gave Alyona k tasks. For each of the tasks two integers l and r are given and Alyona has to answer the following question: if one keeps the rows from l to r inclusive and deletes all others, will the table be sorted in non-decreasing order in at least one column? Formally, does there exist such j that a_{i}, j ≤ a_{i} + 1, j for all i from l to r - 1 inclusive.

Alyona is too small to deal with this task and asks you to help!


-----Input-----

The first line of the input contains two positive integers n and m (1 ≤ n·m ≤ 100 000) — the number of rows and the number of columns in the table respectively. Note that your are given a constraint that bound the product of these two integers, i.e. the number of elements in the table.

Each of the following n lines contains m integers. The j-th integers in the i of these lines stands for a_{i}, j (1 ≤ a_{i}, j ≤ 10^9).

The next line of the input contains an integer k (1 ≤ k ≤ 100 000) — the number of task that teacher gave to Alyona.

The i-th of the next k lines contains two integers l_{i} and r_{i} (1 ≤ l_{i} ≤ r_{i} ≤ n).


-----Output-----

Print "Yes" to the i-th line of the output if the table consisting of rows from l_{i} to r_{i} inclusive is sorted in non-decreasing order in at least one column. Otherwise, print "No".


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

Output
Yes
No
Yes
Yes
Yes
No



-----Note-----

In the sample, the whole table is not sorted in any column. However, rows 1–3 are sorted in column 1, while rows 4–5 are sorted in column 3.
user: By the age of three Smart Beaver mastered all arithmetic operations and got this summer homework from the amazed teacher:

You are given a sequence of integers a_1, a_2, ..., a_{n}. Your task is to perform on it m consecutive operations of the following type:  For given numbers x_{i} and v_{i} assign value v_{i} to element a_{x}_{i}.  For given numbers l_{i} and r_{i} you've got to calculate sum $\sum_{x = 0}^{r_{i} - l_{i}}(f_{x} \cdot a_{l_{i} + x})$, where f_0 = f_1 = 1 and at i ≥ 2: f_{i} = f_{i} - 1 + f_{i} - 2.  For a group of three numbers l_{i} r_{i} d_{i} you should increase value a_{x} by d_{i} for all x (l_{i} ≤ x ≤ r_{i}). 

Smart Beaver planned a tour around great Canadian lakes, so he asked you to help him solve the given problem.


-----Input-----

The first line contains two integers n and m (1 ≤ n, m ≤ 2·10^5) — the number of integers in the sequence and the number of operations, correspondingly. The second line contains n integers a_1, a_2, ..., a_{n} (0 ≤ a_{i} ≤ 10^5). Then follow m lines, each describes an operation. Each line starts with an integer t_{i} (1 ≤ t_{i} ≤ 3) — the operation type:   if t_{i} = 1, then next follow two integers x_{i} v_{i} (1 ≤ x_{i} ≤ n, 0 ≤ v_{i} ≤ 10^5);  if t_{i} = 2, then next follow two integers l_{i} r_{i} (1 ≤ l_{i} ≤ r_{i} ≤ n);  if t_{i} = 3, then next follow three integers l_{i} r_{i} d_{i} (1 ≤ l_{i} ≤ r_{i} ≤ n, 0 ≤ d_{i} ≤ 10^5). 

The input limits for scoring 30 points are (subproblem E1):   It is guaranteed that n does not exceed 100, m does not exceed 10000 and there will be no queries of the 3-rd type. 

The input limits for scoring 70 points are (subproblems E1+E2):   It is guaranteed that there will be queries of the 1-st and 2-nd type only. 

The input limits for scoring 100 points are (subproblems E1+E2+E3):   No extra limitations. 


-----Output-----

For each query print the calculated sum modulo 1000000000 (10^9).


-----Examples-----
Input
5 5
1 3 1 2 4
2 1 4
2 1 5
2 2 4
1 3 10
2 1 5

Output
12
32
8
50

Input
5 4
1 3 1 2 4
3 1 4 1
2 2 4
1 2 10
2 1 5

Output
12
45
user: This is the hard version of the problem. The difference between the versions is that in the easy version all prices $a_i$ are different. You can make hacks if and only if you solved both versions of the problem.

Today is Sage's birthday, and she will go shopping to buy ice spheres. All $n$ ice spheres are placed in a row and they are numbered from $1$ to $n$ from left to right. Each ice sphere has a positive integer price. In this version, some prices can be equal.

An ice sphere is cheap if it costs strictly less than two neighboring ice spheres: the nearest to the left and the nearest to the right. The leftmost and the rightmost ice spheres are not cheap. Sage will choose all cheap ice spheres and then buy only them.

You can visit the shop before Sage and reorder the ice spheres as you wish. Find out the maximum number of ice spheres that Sage can buy, and show how the ice spheres should be reordered.


-----Input-----

The first line contains a single integer $n$ $(1 \le n \le 10^5)$ — the number of ice spheres in the shop.

The second line contains $n$ integers $a_1, a_2, \dots, a_n$ $(1 \le a_i \le 10^9)$ — the prices of ice spheres.


-----Output-----

In the first line print the maximum number of ice spheres that Sage can buy.

In the second line print the prices of ice spheres in the optimal order. If there are several correct answers, you can print any of them.


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

Output
3
3 1 4 2 4 2 5 


-----Note-----

In the sample it's not possible to place the ice spheres in any order so that Sage would buy $4$ of them. If the spheres are placed in the order $(3, 1, 4, 2, 4, 2, 5)$, then Sage will buy one sphere for $1$ and two spheres for $2$ each.
user: Given an input string s, reverse the order of the words.

A word is defined as a sequence of non-space characters. The words in s will be separated by at least one space.

Return a string of the words in reverse order concatenated by a single space.

Note that s may contain leading or trailing spaces or multiple spaces between two words. The returned string should only have a single space separating the words. Do not include any extra spaces.


Example 1:

Input: s = "the sky is blue"
Output: "blue is sky the"
Example 2:

Input: s = "  hello world  "
Output: "world hello"
Explanation: Your reversed string should not contain leading or trailing spaces.
Example 3:

Input: s = "a good   example"
Output: "example good a"
Explanation: You need to reduce multiple spaces between two words to a single space in the reversed string.
Example 4:

Input: s = "  Bob    Loves  Alice   "
Output: "Alice Loves Bob"
Example 5:

Input: s = "Alice does not even like bob"
Output: "bob like even not does Alice"
 

Constraints:

1 <= s.length <= 104
s contains English letters (upper-case and lower-case), digits, and spaces ' '.
There is at least one word in s.

Answer: {"fn_name": "reverseWords", "inputs": [["the sky is blue"], ["  hello world  "], ["a good   example"], ["  Bob    Loves  Alice   "], ["Alice does not even like bob"]], "outputs": ["blue is sky the", "world hello", "example good a", "Alice Loves Bob", "bob like even not does Alice"]}
user: There are N people standing on the x-axis.
Let the coordinate of Person i be x_i.
For every i, x_i is an integer between 0 and 10^9 (inclusive).
It is possible that more than one person is standing at the same coordinate.
You will given M pieces of information regarding the positions of these people.
The i-th piece of information has the form (L_i, R_i, D_i).
This means that Person R_i is to the right of Person L_i by D_i units of distance, that is, x_{R_i} - x_{L_i} = D_i holds.
It turns out that some of these M pieces of information may be incorrect.
Determine if there exists a set of values (x_1, x_2, ..., x_N) that is consistent with the given pieces of information.

-----Constraints-----
 - 1 \leq N \leq 100 000
 - 0 \leq M \leq 200 000
 - 1 \leq L_i, R_i \leq N (1 \leq i \leq M)
 - 0 \leq D_i \leq 10 000 (1 \leq i \leq M)
 - L_i \neq R_i (1 \leq i \leq M)
 - If i \neq j, then (L_i, R_i) \neq (L_j, R_j) and (L_i, R_i) \neq (R_j, L_j).
 - D_i are integers.

-----Input-----
Input is given from Standard Input in the following format:
N M
L_1 R_1 D_1
L_2 R_2 D_2
:
L_M R_M D_M

-----Output-----
If there exists a set of values (x_1, x_2, ..., x_N) that is consistent with all given pieces of information, print Yes; if it does not exist, print No.

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

-----Sample Output-----
Yes

Some possible sets of values (x_1, x_2, x_3) are (0, 1, 2) and (101, 102, 103).
user: We have a tree with N vertices numbered 1 to N. The i-th edge in this tree connects Vertex a_i and b_i.
Additionally, each vertex is painted in a color, and the color of Vertex i is c_i. Here, the color of each vertex is represented by an integer between 1 and N (inclusive). The same integer corresponds to the same color; different integers correspond to different colors.
For each k=1, 2, ..., N, solve the following problem:
 - Find the number of simple paths that visit a vertex painted in the color k one or more times.
Note: The simple paths from Vertex u to v and from v to u are not distinguished.

-----Constraints-----
 - 1 \leq N \leq 2 \times 10^5
 - 1 \leq c_i \leq N
 - 1 \leq a_i,b_i \leq N
 - The given graph is a tree.
 - All values in input are integers.

-----Input-----
Input is given from Standard Input in the following format:
N
c_1 c_2 ... c_N
a_1 b_1
:
a_{N-1} b_{N-1}

-----Output-----
Print the answers for k = 1, 2, ..., N in order, each in its own line.

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

-----Sample Output-----
5
4
0

Let P_{i,j} denote the simple path connecting Vertex i and j.
There are 5 simple paths that visit a vertex painted in the color 1 one or more times:
P_{1,1}\,,\,P_{1,2}\,,\,P_{1,3}\,,\,P_{2,3}\,,\,P_{3,3} 
There are 4 simple paths that visit a vertex painted in the color 2 one or more times:
P_{1,2}\,,\,P_{1,3}\,,\,P_{2,2}\,,\,P_{2,3} 
There are no simple paths that visit a vertex painted in the color 3 one or more times.
user: In the year 2168, AtCoder Inc., which is much larger than now, is starting a limited express train service called AtCoder Express.
In the plan developed by the president Takahashi, the trains will run as follows:
 - A train will run for (t_1 + t_2 + t_3 + ... + t_N) seconds.
 - In the first t_1 seconds, a train must run at a speed of at most v_1 m/s (meters per second). Similarly, in the subsequent t_2 seconds, a train must run at a speed of at most v_2 m/s, and so on.
According to the specifications of the trains, the acceleration of a train must be always within ±1m/s^2. Additionally, a train must stop at the beginning and the end of the run.
Find the maximum possible distance that a train can cover in the run.

-----Constraints-----
 - 1 \leq N \leq 100
 - 1 \leq t_i \leq 200
 - 1 \leq v_i \leq 100
 - All input values are integers.

-----Input-----
Input is given from Standard Input in the following format:
N
t_1 t_2 t_3 … t_N
v_1 v_2 v_3 … v_N

-----Output-----
Print the maximum possible that a train can cover in the run.

Output is considered correct if its absolute difference from the judge's output is at most 10^{-3}.

-----Sample Input-----
1
100
30

-----Sample Output-----
2100.000000000000000


The maximum distance is achieved when a train runs as follows:
 - In the first 30 seconds, it accelerates at a rate of 1m/s^2, covering 450 meters.
 - In the subsequent 40 seconds, it maintains the velocity of 30m/s, covering 1200 meters.
 - In the last 30 seconds, it decelerates at the acceleration of -1m/s^2, covering 450 meters.
The total distance covered is 450 + 1200 + 450 = 2100 meters.