meetforever/GRPO_CL huggingface dataset:
user: Recently Vasya decided to improve his pistol shooting skills. Today his coach offered him the following exercise. He placed $n$ cans in a row on a table. Cans are numbered from left to right from $1$ to $n$. Vasya has to knock down each can exactly once to finish the exercise. He is allowed to choose the order in which he will knock the cans down.

Vasya knows that the durability of the $i$-th can is $a_i$. It means that if Vasya has already knocked $x$ cans down and is now about to start shooting the $i$-th one, he will need $(a_i \cdot x + 1)$ shots to knock it down. You can assume that if Vasya starts shooting the $i$-th can, he will be shooting it until he knocks it down.

Your task is to choose such an order of shooting so that the number of shots required to knock each of the $n$ given cans down exactly once is minimum possible.


-----Input-----

The first line of the input contains one integer $n$ $(2 \le n \le 1\,000)$ — the number of cans.

The second line of the input contains the sequence $a_1, a_2, \dots, a_n$ $(1 \le a_i \le 1\,000)$, where $a_i$ is the durability of the $i$-th can.


-----Output-----

In the first line print the minimum number of shots required to knock each of the $n$ given cans down exactly once.

In the second line print the sequence consisting of $n$ distinct integers from $1$ to $n$ — the order of indices of cans that minimizes the number of shots required. If there are several answers, you can print any of them.


-----Examples-----
Input
3
20 10 20

Output
43
1 3 2 

Input
4
10 10 10 10

Output
64
2 1 4 3 

Input
6
5 4 5 4 4 5

Output
69
6 1 3 5 2 4 

Input
2
1 4

Output
3
2 1 



-----Note-----

In the first example Vasya can start shooting from the first can. He knocks it down with the first shot because he haven't knocked any other cans down before. After that he has to shoot the third can. To knock it down he shoots $20 \cdot 1 + 1 = 21$ times. After that only second can remains. To knock it down Vasya shoots $10 \cdot 2 + 1 = 21$ times. So the total number of shots is $1 + 21 + 21 = 43$.

In the second example the order of shooting does not matter because all cans have the same durability.
user: Given is a permutation P_1, \ldots, P_N of 1, \ldots, N.
Find the number of integers i (1 \leq i \leq N) that satisfy the following condition:  
 - For any integer j (1 \leq j \leq i), P_i \leq P_j.

-----Constraints-----
 - 1 \leq N \leq 2 \times 10^5
 - P_1, \ldots, P_N is a permutation of 1, \ldots, N.  
 - All values in input are integers.

-----Input-----
Input is given from Standard Input in the following format:
N
P_1 ... P_N

-----Output-----
Print the number of integers i that satisfy the condition.

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

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

i=1, 2, and 4 satisfy the condition, but i=3 does not - for example, P_i > P_j holds for j = 1.

Similarly, i=5 does not satisfy the condition, either. Thus, there are three integers that satisfy the condition.
user: You are given a sequence of positive integers of length N, a = (a_1, a_2, ..., a_N).
Your objective is to remove some of the elements in a so that a will be a good sequence.
Here, an sequence b is a good sequence when the following condition holds true:
 - For each element x in b, the value x occurs exactly x times in b.
For example, (3, 3, 3), (4, 2, 4, 1, 4, 2, 4) and () (an empty sequence) are good sequences, while (3, 3, 3, 3) and (2, 4, 1, 4, 2) are not.
Find the minimum number of elements that needs to be removed so that a will be a good sequence.

-----Constraints-----
 - 1 \leq N \leq 10^5
 - a_i is an integer.
 - 1 \leq a_i \leq 10^9

-----Input-----
Input is given from Standard Input in the following format:
N
a_1 a_2 ... a_N

-----Output-----
Print the minimum number of elements that needs to be removed so that a will be a good sequence.

-----Sample Input-----
4
3 3 3 3

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

We can, for example, remove one occurrence of 3. Then, (3, 3, 3) is a good sequence.
user: We have five variables x_1, x_2, x_3, x_4, and x_5.
The variable x_i was initially assigned a value of i.
Snuke chose one of these variables and assigned it 0.
You are given the values of the five variables after this assignment.
Find out which variable Snuke assigned 0.

-----Constraints-----
 - The values of x_1, x_2, x_3, x_4, and x_5 given as input are a possible outcome of the assignment by Snuke.

-----Input-----
Input is given from Standard Input in the following format:
x_1 x_2 x_3 x_4 x_5

-----Output-----
If the variable Snuke assigned 0 was x_i, print the integer i.

-----Sample Input-----
0 2 3 4 5

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

In this case, Snuke assigned 0 to x_1, so we should print 1.
user: There is a bar of chocolate with a height of H blocks and a width of W blocks.
Snuke is dividing this bar into exactly three pieces.
He can only cut the bar along borders of blocks, and the shape of each piece must be a rectangle.
Snuke is trying to divide the bar as evenly as possible.
More specifically, he is trying to minimize S_{max} - S_{min}, where S_{max} is the area (the number of blocks contained) of the largest piece, and S_{min} is the area of the smallest piece.
Find the minimum possible value of S_{max} - S_{min}.

-----Constraints-----
 - 2 ≤ H, W ≤ 10^5

-----Input-----
Input is given from Standard Input in the following format:
H W

-----Output-----
Print the minimum possible value of S_{max} - S_{min}.

-----Sample Input-----
3 5

-----Sample Output-----
0

In the division below, S_{max} - S_{min} = 5 - 5 = 0.
user: We have a sequence of length N, a = (a_1, a_2, ..., a_N).
Each a_i is a positive integer.
Snuke's objective is to permute the element in a so that the following condition is satisfied:
 - For each 1 ≤ i ≤ N - 1, the product of a_i and a_{i + 1} is a multiple of 4.
Determine whether Snuke can achieve his objective.

-----Constraints-----
 - 2 ≤ N ≤ 10^5
 - a_i is an integer.
 - 1 ≤ a_i ≤ 10^9

-----Input-----
Input is given from Standard Input in the following format:
N
a_1 a_2 ... a_N

-----Output-----
If Snuke can achieve his objective, print Yes; otherwise, print No.

-----Sample Input-----
3
1 10 100

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

One solution is (1, 100, 10).
user: You are given strings s and t, consisting of lowercase English letters.
You will create a string s' by freely rearranging the characters in s.
You will also create a string t' by freely rearranging the characters in t.
Determine whether it is possible to satisfy s' < t' for the lexicographic order.

-----Notes-----
For a string a = a_1 a_2 ... a_N of length N and a string b = b_1 b_2 ... b_M of length M, we say a < b for the lexicographic order if either one of the following two conditions holds true:
 - N < M and a_1 = b_1, a_2 = b_2, ..., a_N = b_N.
 - There exists i (1 \leq i \leq N, M) such that a_1 = b_1, a_2 = b_2, ..., a_{i - 1} = b_{i - 1} and a_i < b_i. Here, letters are compared using alphabetical order.
For example, xy < xya and atcoder < atlas.

-----Constraints-----
 - The lengths of s and t are between 1 and 100 (inclusive).
 - s and t consists of lowercase English letters.

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

-----Output-----
If it is possible to satisfy s' < t', print Yes; if it is not, print No.

-----Sample Input-----
yx
axy

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

We can, for example, rearrange yx into xy and axy into yxa. Then, xy < yxa.
user: We ask you to select some number of positive integers, and calculate the sum of them.
It is allowed to select as many integers as you like, and as large integers as you wish.
You have to follow these, however: each selected integer needs to be a multiple of A, and you need to select at least one integer.
Your objective is to make the sum congruent to C modulo B.
Determine whether this is possible.
If the objective is achievable, print YES. Otherwise, print NO.

-----Constraints-----
 - 1 ≤ A ≤ 100
 - 1 ≤ B ≤ 100
 - 0 ≤ C < B

-----Input-----
Input is given from Standard Input in the following format:
A B C

-----Output-----
Print YES or NO.

-----Sample Input-----
7 5 1

-----Sample Output-----
YES

For example, if you select 7 and 14, the sum 21 is congruent to 1 modulo 5.
user: There is a farm whose length and width are A yard and B yard, respectively. A farmer, John, made a vertical road and a horizontal road inside the farm from one border to another, as shown below: (The gray part represents the roads.)

What is the area of this yard excluding the roads? Find it.

-----Note-----
It can be proved that the positions of the roads do not affect the area.

-----Constraints-----
 - A is an integer between 2 and 100 (inclusive).
 - B is an integer between 2 and 100 (inclusive).

-----Input-----
Input is given from Standard Input in the following format:
A B

-----Output-----
Print the area of this yard excluding the roads (in square yards).

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

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

In this case, the area is 1 square yard.
user: We have a long seat of width X centimeters.
There are many people who wants to sit here. A person sitting on the seat will always occupy an interval of length Y centimeters.
We would like to seat as many people as possible, but they are all very shy, and there must be a gap of length at least Z centimeters between two people, and between the end of the seat and a person.
At most how many people can sit on the seat?

-----Constraints-----
 - All input values are integers.
 - 1 \leq X, Y, Z \leq 10^5
 - Y+2Z \leq X

-----Input-----
Input is given from Standard Input in the following format:
X Y Z

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

-----Sample Input-----
13 3 1

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

There is just enough room for three, as shown below:
Figure
user: On a two-dimensional plane, there are N red points and N blue points.
The coordinates of the i-th red point are (a_i, b_i), and the coordinates of the i-th blue point are (c_i, d_i).
A red point and a blue point can form a friendly pair when, the x-coordinate of the red point is smaller than that of the blue point, and the y-coordinate of the red point is also smaller than that of the blue point.
At most how many friendly pairs can you form? Note that a point cannot belong to multiple pairs.

-----Constraints-----
 - All input values are integers.
 - 1 \leq N \leq 100
 - 0 \leq a_i, b_i, c_i, d_i < 2N
 - a_1, a_2, ..., a_N, c_1, c_2, ..., c_N are all different.
 - b_1, b_2, ..., b_N, d_1, d_2, ..., d_N are all different.

-----Input-----
Input is given from Standard Input in the following format:
N
a_1 b_1
a_2 b_2
:
a_N b_N
c_1 d_1
c_2 d_2
:
c_N d_N

-----Output-----
Print the maximum number of friendly pairs.

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

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

For example, you can pair (2, 0) and (4, 2), then (3, 1) and (5, 5).
user: In a public bath, there is a shower which emits water for T seconds when the switch is pushed.
If the switch is pushed when the shower is already emitting water, from that moment it will be emitting water for T seconds.
Note that it does not mean that the shower emits water for T additional seconds.
N people will push the switch while passing by the shower.
The i-th person will push the switch t_i seconds after the first person pushes it.
How long will the shower emit water in total?

-----Constraints-----
 - 1 ≤ N ≤ 200,000
 - 1 ≤ T ≤ 10^9
 - 0 = t_1 < t_2 < t_3 < , ..., < t_{N-1} < t_N ≤ 10^9
 - T and each t_i are integers.

-----Input-----
Input is given from Standard Input in the following format:
N T
t_1 t_2 ... t_N

-----Output-----
Assume that the shower will emit water for a total of X seconds. Print X.

-----Sample Input-----
2 4
0 3

-----Sample Output-----
7

Three seconds after the first person pushes the water, the switch is pushed again and the shower emits water for four more seconds, for a total of seven seconds.
user: You have got a shelf and want to put some books on it.

You are given $q$ queries of three types:  L $id$ — put a book having index $id$ on the shelf to the left from the leftmost existing book;  R $id$ — put a book having index $id$ on the shelf to the right from the rightmost existing book;  ? $id$ — calculate the minimum number of books you need to pop from the left or from the right in such a way that the book with index $id$ will be leftmost or rightmost. 

You can assume that the first book you will put can have any position (it does not matter) and queries of type $3$ are always valid (it is guaranteed that the book in each such query is already placed). You can also assume that you don't put the same book on the shelf twice, so $id$s don't repeat in queries of first two types.

Your problem is to answer all the queries of type $3$ in order they appear in the input.

Note that after answering the query of type $3$ all the books remain on the shelf and the relative order of books does not change.

If you are Python programmer, consider using PyPy instead of Python when you submit your code.


-----Input-----

The first line of the input contains one integer $q$ ($1 \le q \le 2 \cdot 10^5$) — the number of queries.

Then $q$ lines follow. The $i$-th line contains the $i$-th query in format as in the problem statement. It is guaranteed that queries are always valid (for query type $3$, it is guaranteed that the book in each such query is already placed, and for other types, it is guaranteed that the book was not placed before).

It is guaranteed that there is at least one query of type $3$ in the input.

In each query the constraint $1 \le id \le 2 \cdot 10^5$ is met.


-----Output-----

Print answers to queries of the type $3$ in order they appear in the input.


-----Examples-----
Input
8
L 1
R 2
R 3
? 2
L 4
? 1
L 5
? 1

Output
1
1
2

Input
10
L 100
R 100000
R 123
L 101
? 123
L 10
R 115
? 100
R 110
? 115

Output
0
2
1



-----Note-----

Let's take a look at the first example and let's consider queries:   The shelf will look like $[1]$;  The shelf will look like $[1, 2]$;  The shelf will look like $[1, 2, 3]$;  The shelf looks like $[1, \textbf{2}, 3]$ so the answer is $1$;  The shelf will look like $[4, 1, 2, 3]$;  The shelf looks like $[4, \textbf{1}, 2, 3]$ so the answer is $1$;  The shelf will look like $[5, 4, 1, 2, 3]$;  The shelf looks like $[5, 4, \textbf{1}, 2, 3]$ so the answer is $2$. 

Let's take a look at the second example and let's consider queries:   The shelf will look like $[100]$;  The shelf will look like $[100, 100000]$;  The shelf will look like $[100, 100000, 123]$;  The shelf will look like $[101, 100, 100000, 123]$;  The shelf looks like $[101, 100, 100000, \textbf{123}]$ so the answer is $0$;  The shelf will look like $[10, 101, 100, 100000, 123]$;  The shelf will look like $[10, 101, 100, 100000, 123, 115]$;  The shelf looks like $[10, 101, \textbf{100}, 100000, 123, 115]$ so the answer is $2$;  The shelf will look like $[10, 101, 100, 100000, 123, 115, 110]$;  The shelf looks like $[10, 101, 100, 100000, 123, \textbf{115}, 110]$ so the answer is $1$.
user: You are given an integer $n$.

You can perform any of the following operations with this number an arbitrary (possibly, zero) number of times:   Replace $n$ with $\frac{n}{2}$ if $n$ is divisible by $2$;  Replace $n$ with $\frac{2n}{3}$ if $n$ is divisible by $3$;  Replace $n$ with $\frac{4n}{5}$ if $n$ is divisible by $5$. 

For example, you can replace $30$ with $15$ using the first operation, with $20$ using the second operation or with $24$ using the third operation.

Your task is to find the minimum number of moves required to obtain $1$ from $n$ or say that it is impossible to do it.

You have to answer $q$ independent queries.


-----Input-----

The first line of the input contains one integer $q$ ($1 \le q \le 1000$) — the number of queries.

The next $q$ lines contain the queries. For each query you are given the integer number $n$ ($1 \le n \le 10^{18}$).


-----Output-----

Print the answer for each query on a new line. If it is impossible to obtain $1$ from $n$, print -1. Otherwise, print the minimum number of moves required to do it.


-----Example-----
Input
7
1
10
25
30
14
27
1000000000000000000

Output
0
4
6
6
-1
6
72
user: You are given some Tetris field consisting of $n$ columns. The initial height of the $i$-th column of the field is $a_i$ blocks. On top of these columns you can place only figures of size $2 \times 1$ (i.e. the height of this figure is $2$ blocks and the width of this figure is $1$ block). Note that you cannot rotate these figures.

Your task is to say if you can clear the whole field by placing such figures.

More formally, the problem can be described like this:

The following process occurs while at least one $a_i$ is greater than $0$:  You place one figure $2 \times 1$ (choose some $i$ from $1$ to $n$ and replace $a_i$ with $a_i + 2$);  then, while all $a_i$ are greater than zero, replace each $a_i$ with $a_i - 1$. 

And your task is to determine if it is possible to clear the whole field (i.e. finish the described process), choosing the places for new figures properly.

You have to answer $t$ independent test cases.


-----Input-----

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

The next $2t$ lines describe test cases. The first line of the test case contains one integer $n$ ($1 \le n \le 100$) — the number of columns in the Tetris field. The second line of the test case contains $n$ integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 100$), where $a_i$ is the initial height of the $i$-th column of the Tetris field.


-----Output-----

For each test case, print the answer — "YES" (without quotes) if you can clear the whole Tetris field and "NO" otherwise.


-----Example-----
Input
4
3
1 1 3
4
1 1 2 1
2
11 11
1
100

Output
YES
NO
YES
YES



-----Note-----

The first test case of the example field is shown below:

[Image]

Gray lines are bounds of the Tetris field. Note that the field has no upper bound.

One of the correct answers is to first place the figure in the first column. Then after the second step of the process, the field becomes $[2, 0, 2]$. Then place the figure in the second column and after the second step of the process, the field becomes $[0, 0, 0]$.

And the second test case of the example field is shown below:

$\square$

It can be shown that you cannot do anything to end the process.

In the third test case of the example, you first place the figure in the second column after the second step of the process, the field becomes $[0, 2]$. Then place the figure in the first column and after the second step of the process, the field becomes $[0, 0]$.

In the fourth test case of the example, place the figure in the first column, then the field becomes $[102]$ after the first step of the process, and then the field becomes $[0]$ after the second step of the process.
user: You are given two strings $a$ and $b$ consisting of lowercase English letters, both of length $n$. The characters of both strings have indices from $1$ to $n$, inclusive. 

You are allowed to do the following changes:   Choose any index $i$ ($1 \le i \le n$) and swap characters $a_i$ and $b_i$;  Choose any index $i$ ($1 \le i \le n$) and swap characters $a_i$ and $a_{n - i + 1}$;  Choose any index $i$ ($1 \le i \le n$) and swap characters $b_i$ and $b_{n - i + 1}$. 

Note that if $n$ is odd, you are formally allowed to swap $a_{\lceil\frac{n}{2}\rceil}$ with $a_{\lceil\frac{n}{2}\rceil}$ (and the same with the string $b$) but this move is useless. Also you can swap two equal characters but this operation is useless as well.

You have to make these strings equal by applying any number of changes described above, in any order. But it is obvious that it may be impossible to make two strings equal by these swaps.

In one preprocess move you can replace a character in $a$ with another character. In other words, in a single preprocess move you can choose any index $i$ ($1 \le i \le n$), any character $c$ and set $a_i := c$.

Your task is to find the minimum number of preprocess moves to apply in such a way that after them you can make strings $a$ and $b$ equal by applying some number of changes described in the list above.

Note that the number of changes you make after the preprocess moves does not matter. Also note that you cannot apply preprocess moves to the string $b$ or make any preprocess moves after the first change is made.


-----Input-----

The first line of the input contains one integer $n$ ($1 \le n \le 10^5$) — the length of strings $a$ and $b$.

The second line contains the string $a$ consisting of exactly $n$ lowercase English letters.

The third line contains the string $b$ consisting of exactly $n$ lowercase English letters.


-----Output-----

Print a single integer — the minimum number of preprocess moves to apply before changes, so that it is possible to make the string $a$ equal to string $b$ with a sequence of changes from the list above.


-----Examples-----
Input
7
abacaba
bacabaa

Output
4

Input
5
zcabd
dbacz

Output
0



-----Note-----

In the first example preprocess moves are as follows: $a_1 := $'b', $a_3 := $'c', $a_4 := $'a' and $a_5:=$'b'. Afterwards, $a = $"bbcabba". Then we can obtain equal strings by the following sequence of changes: $swap(a_2, b_2)$ and $swap(a_2, a_6)$. There is no way to use fewer than $4$ preprocess moves before a sequence of changes to make string equal, so the answer in this example is $4$.

In the second example no preprocess moves are required. We can use the following sequence of changes to make $a$ and $b$ equal: $swap(b_1, b_5)$, $swap(a_2, a_4)$.
user: A frog is currently at the point $0$ on a coordinate axis $Ox$. It jumps by the following algorithm: the first jump is $a$ units to the right, the second jump is $b$ units to the left, the third jump is $a$ units to the right, the fourth jump is $b$ units to the left, and so on.

Formally:   if the frog has jumped an even number of times (before the current jump), it jumps from its current position $x$ to position $x+a$;  otherwise it jumps from its current position $x$ to position $x-b$. 

Your task is to calculate the position of the frog after $k$ jumps.

But... One more thing. You are watching $t$ different frogs so you have to answer $t$ independent queries.


-----Input-----

The first line of the input contains one integer $t$ ($1 \le t \le 1000$) — the number of queries.

Each of the next $t$ lines contain queries (one query per line).

The query is described as three space-separated integers $a, b, k$ ($1 \le a, b, k \le 10^9$) — the lengths of two types of jumps and the number of jumps, respectively.


-----Output-----

Print $t$ integers. The $i$-th integer should be the answer for the $i$-th query.


-----Example-----
Input
6
5 2 3
100 1 4
1 10 5
1000000000 1 6
1 1 1000000000
1 1 999999999

Output
8
198
-17
2999999997
0
1



-----Note-----

In the first query frog jumps $5$ to the right, $2$ to the left and $5$ to the right so the answer is $5 - 2 + 5 = 8$.

In the second query frog jumps $100$ to the right, $1$ to the left, $100$ to the right and $1$ to the left so the answer is $100 - 1 + 100 - 1 = 198$.

In the third query the answer is $1 - 10 + 1 - 10 + 1 = -17$.

In the fourth query the answer is $10^9 - 1 + 10^9 - 1 + 10^9 - 1 = 2999999997$.

In the fifth query all frog's jumps are neutralized by each other so the answer is $0$.

The sixth query is the same as the fifth but without the last jump so the answer is $1$.