meetforever/GRPO_CL huggingface dataset:
user: This problem is different from the easy version. In this version Ujan makes at most $2n$ swaps. In addition, $k \le 1000, n \le 50$ and it is necessary to print swaps themselves. You can hack this problem if you solve it. But you can hack the previous problem only if you solve both problems.

After struggling and failing many times, Ujan decided to try to clean up his house again. He decided to get his strings in order first.

Ujan has two distinct strings $s$ and $t$ of length $n$ consisting of only of lowercase English characters. He wants to make them equal. Since Ujan is lazy, he will perform the following operation at most $2n$ times: he takes two positions $i$ and $j$ ($1 \le i,j \le n$, the values $i$ and $j$ can be equal or different), and swaps the characters $s_i$ and $t_j$.

Ujan's goal is to make the strings $s$ and $t$ equal. He does not need to minimize the number of performed operations: any sequence of operations of length $2n$ or shorter is suitable.


-----Input-----

The first line contains a single integer $k$ ($1 \leq k \leq 1000$), the number of test cases.

For each of the test cases, the first line contains a single integer $n$ ($2 \leq n \leq 50$), the length of the strings $s$ and $t$. 

Each of the next two lines contains the strings $s$ and $t$, each having length exactly $n$. The strings consist only of lowercase English letters. It is guaranteed that strings are different.


-----Output-----

For each test case, output "Yes" if Ujan can make the two strings equal with at most $2n$ operations and "No" otherwise. You can print each letter in any case (upper or lower).

In the case of "Yes" print $m$ ($1 \le m \le 2n$) on the next line, where $m$ is the number of swap operations to make the strings equal. Then print $m$ lines, each line should contain two integers $i, j$ ($1 \le i, j \le n$) meaning that Ujan swaps $s_i$ and $t_j$ during the corresponding operation. You do not need to minimize the number of operations. Any sequence of length not more than $2n$ is suitable.


-----Example-----
Input
4
5
souse
houhe
3
cat
dog
2
aa
az
3
abc
bca

Output
Yes
1
1 4
No
No
Yes
3
1 2
3 1
2 3
user: You have a string $s$ consisting of $n$ characters. Each character is either 0 or 1.

You can perform operations on the string. Each operation consists of two steps:  select an integer $i$ from $1$ to the length of the string $s$, then delete the character $s_i$ (the string length gets reduced by $1$, the indices of characters to the right of the deleted one also get reduced by $1$);  if the string $s$ is not empty, delete the maximum length prefix consisting of the same characters (the indices of the remaining characters and the string length get reduced by the length of the deleted prefix). 

Note that both steps are mandatory in each operation, and their order cannot be changed.

For example, if you have a string $s =$ 111010, the first operation can be one of the following:  select $i = 1$: we'll get 111010 $\rightarrow$ 11010 $\rightarrow$ 010;  select $i = 2$: we'll get 111010 $\rightarrow$ 11010 $\rightarrow$ 010;  select $i = 3$: we'll get 111010 $\rightarrow$ 11010 $\rightarrow$ 010;  select $i = 4$: we'll get 111010 $\rightarrow$ 11110 $\rightarrow$ 0;  select $i = 5$: we'll get 111010 $\rightarrow$ 11100 $\rightarrow$ 00;  select $i = 6$: we'll get 111010 $\rightarrow$ 11101 $\rightarrow$ 01. 

You finish performing operations when the string $s$ becomes empty. What is the maximum number of operations you can perform?


-----Input-----

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

The first line of each test case contains a single integer $n$ ($1 \le n \le 2 \cdot 10^5$) — the length of the string $s$.

The second line contains string $s$ of $n$ characters. Each character is either 0 or 1.

It's guaranteed that the total sum of $n$ over test cases doesn't exceed $2 \cdot 10^5$.


-----Output-----

For each test case, print a single integer — the maximum number of operations you can perform.


-----Example-----
Input
5
6
111010
1
0
1
1
2
11
6
101010

Output
3
1
1
1
3



-----Note-----

In the first test case, you can, for example, select $i = 2$ and get string 010 after the first operation. After that, you can select $i = 3$ and get string 1. Finally, you can only select $i = 1$ and get empty string.
user: Bertown is a city with $n$ buildings in a straight line.

The city's security service discovered that some buildings were mined. A map was compiled, which is a string of length $n$, where the $i$-th character is "1" if there is a mine under the building number $i$ and "0" otherwise.

Bertown's best sapper knows how to activate mines so that the buildings above them are not damaged. When a mine under the building numbered $x$ is activated, it explodes and activates two adjacent mines under the buildings numbered $x-1$ and $x+1$ (if there were no mines under the building, then nothing happens). Thus, it is enough to activate any one mine on a continuous segment of mines to activate all the mines of this segment. For manual activation of one mine, the sapper takes $a$ coins. He can repeat this operation as many times as you want.

Also, a sapper can place a mine under a building if it wasn't there. For such an operation, he takes $b$ coins. He can also repeat this operation as many times as you want.

The sapper can carry out operations in any order.

You want to blow up all the mines in the city to make it safe. Find the minimum number of coins that the sapper will have to pay so that after his actions there are no mines left in the city.


-----Input-----

The first line contains one positive integer $t$ ($1 \le t \le 10^5$) — the number of test cases. Then $t$ test cases follow.

Each test case begins with a line containing two integers $a$ and $b$ ($1 \le a, b \le 1000$) — the cost of activating and placing one mine, respectively.

The next line contains a map of mines in the city — a string consisting of zeros and ones.

The sum of the string lengths for all test cases does not exceed $10^5$.


-----Output-----

For each test case, output one integer — the minimum number of coins that the sapper will have to pay.


-----Example-----
Input
2
1 1
01000010
5 1
01101110

Output
2
6



-----Note-----

In the second test case, if we place a mine under the fourth building and then activate it, then all mines on the field are activated. The cost of such operations is six, $b=1$ coin for placing a mine and $a=5$ coins for activating.
user: Word $s$ of length $n$ is called $k$-complete if   $s$ is a palindrome, i.e. $s_i=s_{n+1-i}$ for all $1 \le i \le n$;  $s$ has a period of $k$, i.e. $s_i=s_{k+i}$ for all $1 \le i \le n-k$. 

For example, "abaaba" is a $3$-complete word, while "abccba" is not.

Bob is given a word $s$ of length $n$ consisting of only lowercase Latin letters and an integer $k$, such that $n$ is divisible by $k$. He wants to convert $s$ to any $k$-complete word.

To do this Bob can choose some $i$ ($1 \le i \le n$) and replace the letter at position $i$ with some other lowercase Latin letter.

So now Bob wants to know the minimum number of letters he has to replace to convert $s$ to any $k$-complete word.

Note that Bob can do zero changes if the word $s$ is already $k$-complete.

You are required to answer $t$ test cases independently.


-----Input-----

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

The first line of each test case contains two integers $n$ and $k$ ($1 \le k < n \le 2 \cdot 10^5$, $n$ is divisible by $k$).

The second line of each test case contains a word $s$ of length $n$.

It is guaranteed that word $s$ only contains lowercase Latin letters. And it is guaranteed that the sum of $n$ over all test cases will not exceed $2 \cdot 10^5$.


-----Output-----

For each test case, output one integer, representing the minimum number of characters he has to replace to convert $s$ to any $k$-complete word.


-----Example-----
Input
4
6 2
abaaba
6 3
abaaba
36 9
hippopotomonstrosesquippedaliophobia
21 7
wudixiaoxingxingheclp

Output
2
0
23
16



-----Note-----

In the first test case, one optimal solution is aaaaaa.

In the second test case, the given word itself is $k$-complete.
user: You're given an array $a$ of $n$ integers, such that $a_1 + a_2 + \cdots + a_n = 0$.

In one operation, you can choose two different indices $i$ and $j$ ($1 \le i, j \le n$), decrement $a_i$ by one and increment $a_j$ by one. If $i < j$ this operation is free, otherwise it costs one coin.

How many coins do you have to spend in order to make all elements equal to $0$?


-----Input-----

Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 5000$). Description of the test cases follows.

The first line of each test case contains an integer $n$ ($1 \le n \le 10^5$)  — the number of elements.

The next line contains $n$ integers $a_1, \ldots, a_n$ ($-10^9 \le a_i \le 10^9$). It is given that $\sum_{i=1}^n a_i = 0$.

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


-----Output-----

For each test case, print the minimum number of coins we have to spend in order to make all elements equal to $0$.


-----Example-----
Input
7
4
-3 5 -3 1
2
1 -1
4
-3 2 -3 4
4
-1 1 1 -1
7
-5 7 -6 -4 17 -13 4
6
-1000000000 -1000000000 -1000000000 1000000000 1000000000 1000000000
1
0

Output
3
0
4
1
8
3000000000
0



-----Note-----

Possible strategy for the first test case:   Do $(i=2, j=3)$ three times (free), $a = [-3, 2, 0, 1]$.  Do $(i=2, j=1)$ two times (pay two coins), $a = [-1, 0, 0, 1]$.  Do $(i=4, j=1)$ one time (pay one coin), $a = [0, 0, 0, 0]$.
user: Phoenix loves beautiful arrays. An array is beautiful if all its subarrays of length $k$ have the same sum. A subarray of an array is any sequence of consecutive elements.

Phoenix currently has an array $a$ of length $n$. He wants to insert some number of integers, possibly zero, into his array such that it becomes beautiful. The inserted integers must be between $1$ and $n$ inclusive. Integers may be inserted anywhere (even before the first or after the last element), and he is not trying to minimize the number of inserted integers.


-----Input-----

The input consists of multiple test cases. The first line contains an integer $t$ ($1 \le t \le 50$) — the number of test cases.

The first line of each test case contains two integers $n$ and $k$ ($1 \le k \le n \le 100$).

The second line of each test case contains $n$ space-separated integers ($1 \le a_i \le n$) — the array that Phoenix currently has. This array may or may not be already beautiful.


-----Output-----

For each test case, if it is impossible to create a beautiful array, print -1. Otherwise, print two lines.

The first line should contain the length of the beautiful array $m$ ($n \le m \le 10^4$). You don't need to minimize $m$.

The second line should contain $m$ space-separated integers ($1 \le b_i \le n$) — a beautiful array that Phoenix can obtain after inserting some, possibly zero, integers into his array $a$. You may print integers that weren't originally in array $a$.

If there are multiple solutions, print any. It's guaranteed that if we can make array $a$ beautiful, we can always make it with resulting length no more than $10^4$.


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

Output
5
1 2 1 2 1
4
1 2 2 1
-1
7
4 3 2 1 4 3 2


-----Note-----

In the first test case, we can make array $a$ beautiful by inserting the integer $1$ at index $3$ (in between the two existing $2$s). Now, all subarrays of length $k=2$ have the same sum $3$. There exists many other possible solutions, for example:   $2, 1, 2, 1, 2, 1$  $1, 2, 1, 2, 1, 2$ 

In the second test case, the array is already beautiful: all subarrays of length $k=3$ have the same sum $5$.

In the third test case, it can be shown that we cannot insert numbers to make array $a$ beautiful.

In the fourth test case, the array $b$ shown is beautiful and all subarrays of length $k=4$ have the same sum $10$. There exist other solutions also.
user: You're given an array of $n$ integers between $0$ and $n$ inclusive.

In one operation, you can choose any element of the array and replace it by the MEX of the elements of the array (which may change after the operation).

For example, if the current array is $[0, 2, 2, 1, 4]$, you can choose the second element and replace it by the MEX of the present elements  — $3$. Array will become $[0, 3, 2, 1, 4]$.

You must make the array non-decreasing, using at most $2n$ operations.

It can be proven that it is always possible. Please note that you do not have to minimize the number of operations. If there are many solutions, you can print any of them.

 –

An array $b[1 \ldots n]$ is non-decreasing if and only if $b_1 \le b_2 \le \ldots \le b_n$.

The MEX (minimum excluded) of an array is the smallest non-negative integer that does not belong to the array. For instance:  The MEX of $[2, 2, 1]$ is $0$, because $0$ does not belong to the array.  The MEX of $[3, 1, 0, 1]$ is $2$, because $0$ and $1$ belong to the array, but $2$ does not.  The MEX of $[0, 3, 1, 2]$ is $4$ because $0$, $1$, $2$ and $3$ belong to the array, but $4$ does not. 

It's worth mentioning that the MEX of an array of length $n$ is always between $0$ and $n$ inclusive.


-----Input-----

The first line contains a single integer $t$ ($1 \le t \le 200$) — the number of test cases. The description of the test cases follows.

The first line of each test case contains a single integer $n$ ($3 \le n \le 1000$) — length of the array.

The second line of each test case contains $n$ integers $a_1, \ldots, a_n$ ($0 \le a_i \le n$) — elements of the array. Note that they don't have to be distinct.

It is guaranteed that the sum of $n$ over all test cases doesn't exceed $1000$.


-----Output-----

For each test case, you must output two lines:

The first line must contain a single integer $k$ ($0 \le k \le 2n$)  — the number of operations you perform.

The second line must contain $k$ integers $x_1, \ldots, x_k$ ($1 \le x_i \le n$), where $x_i$ is the index chosen for the $i$-th operation.

If there are many solutions, you can find any of them. Please remember that it is not required to minimize $k$.


-----Example-----
Input
5
3
2 2 3
3
2 1 0
7
0 7 3 1 3 7 7
9
2 0 1 1 2 4 4 2 0
9
8 4 7 6 1 2 3 0 5

Output
0

2
3 1
4
2 5 5 4
11
3 8 9 7 8 5 9 6 4 1 2
10
1 8 1 9 5 2 4 6 3 7



-----Note-----

In the first test case, the array is already non-decreasing ($2 \le 2 \le 3$).

Explanation of the second test case (the element modified by each operation is colored in red):   $a = [2, 1, 0]$ ; the initial MEX is $3$.  $a = [2, 1, \color{red}{3}]$ ; the new MEX is $0$.  $a = [\color{red}{0}, 1, 3]$ ; the new MEX is $2$.  The final array is non-decreasing: $0 \le 1 \le 3$. 

Explanation of the third test case:   $a = [0, 7, 3, 1, 3, 7, 7]$ ; the initial MEX is $2$.  $a = [0, \color{red}{2}, 3, 1, 3, 7, 7]$ ; the new MEX is $4$.  $a = [0, 2, 3, 1, \color{red}{4}, 7, 7]$ ; the new MEX is $5$.  $a = [0, 2, 3, 1, \color{red}{5}, 7, 7]$ ; the new MEX is $4$.  $a = [0, 2, 3, \color{red}{4}, 5, 7, 7]$ ; the new MEX is $1$.  The final array is non-decreasing: $0 \le 2 \le 3 \le 4 \le 5 \le 7 \le 7$.
user: Polycarp plays a computer game. In this game, the players summon armies of magical minions, which then fight each other.

Polycarp can summon $n$ different minions. The initial power level of the $i$-th minion is $a_i$, and when it is summoned, all previously summoned minions' power levels are increased by $b_i$. The minions can be summoned in any order.

Unfortunately, Polycarp cannot have more than $k$ minions under his control. To get rid of unwanted minions after summoning them, he may destroy them. Each minion can be summoned (and destroyed) only once.

Polycarp's goal is to summon the strongest possible army. Formally, he wants to maximize the sum of power levels of all minions under his control (those which are summoned and not destroyed).

Help Polycarp to make up a plan of actions to summon the strongest possible army!


-----Input-----

The first line contains one integer $T$ ($1 \le T \le 75$) — the number of test cases.

Each test case begins with a line containing two integers $n$ and $k$ ($1 \le k \le n \le 75$) — the number of minions availible for summoning, and the maximum number of minions that can be controlled by Polycarp, respectively.

Then $n$ lines follow, the $i$-th line contains $2$ integers $a_i$ and $b_i$ ($1 \le a_i \le 10^5$, $0 \le b_i \le 10^5$) — the parameters of the $i$-th minion.


-----Output-----

For each test case print the optimal sequence of actions as follows:

Firstly, print $m$ — the number of actions which Polycarp has to perform ($0 \le m \le 2n$). Then print $m$ integers $o_1$, $o_2$, ..., $o_m$, where $o_i$ denotes the $i$-th action as follows: if the $i$-th action is to summon the minion $x$, then $o_i = x$, and if the $i$-th action is to destroy the minion $x$, then $o_i = -x$. Each minion can be summoned at most once and cannot be destroyed before being summoned (and, obviously, cannot be destroyed more than once). The number of minions in Polycarp's army should be not greater than $k$ after every action.

If there are multiple optimal sequences, print any of them.


-----Example-----
Input
3
5 2
5 3
7 0
5 0
4 0
10 0
2 1
10 100
50 10
5 5
1 5
2 4
3 3
4 2
5 1

Output
4
2 1 -1 5
1
2
5
5 4 3 2 1



-----Note-----

Consider the example test.

In the first test case, Polycarp can summon the minion $2$ with power level $7$, then summon the minion $1$, which will increase the power level of the previous minion by $3$, then destroy the minion $1$, and finally, summon the minion $5$. After this, Polycarp will have two minions with power levels of $10$.

In the second test case, Polycarp can control only one minion, so he should choose the strongest of them and summon it.

In the third test case, Polycarp is able to summon and control all five minions.
user: The statement of this problem is the same as the statement of problem C1. The only difference is that, in problem C1, $n$ is always even, and in C2, $n$ is always odd.

You are given a regular polygon with $2 \cdot n$ vertices (it's convex and has equal sides and equal angles) and all its sides have length $1$. Let's name it as $2n$-gon.

Your task is to find the square of the minimum size such that you can embed $2n$-gon in the square. Embedding $2n$-gon in the square means that you need to place $2n$-gon in the square in such way that each point which lies inside or on a border of $2n$-gon should also lie inside or on a border of the square.

You can rotate $2n$-gon and/or the square.


-----Input-----

The first line contains a single integer $T$ ($1 \le T \le 200$) — the number of test cases.

Next $T$ lines contain descriptions of test cases — one per line. Each line contains single odd integer $n$ ($3 \le n \le 199$). Don't forget you need to embed $2n$-gon, not an $n$-gon.


-----Output-----

Print $T$ real numbers — one per test case. For each test case, print the minimum length of a side of the square $2n$-gon can be embedded in. Your answer will be considered correct if its absolute or relative error doesn't exceed $10^{-6}$.


-----Example-----
Input
3
3
5
199

Output
1.931851653
3.196226611
126.687663595
user: Lee is going to fashionably decorate his house for a party, using some regular convex polygons...

Lee thinks a regular $n$-sided (convex) polygon is beautiful if and only if he can rotate it in such a way that at least one of its edges is parallel to the $OX$-axis and at least one of its edges is parallel to the $OY$-axis at the same time.

Recall that a regular $n$-sided polygon is a convex polygon with $n$ vertices such that all the edges and angles are equal.

Now he is shopping: the market has $t$ regular polygons. For each of them print YES if it is beautiful and NO otherwise.


-----Input-----

The first line contains a single integer $t$ ($1 \le t \le 10^4$) — the number of polygons in the market. 

Each of the next $t$ lines contains a single integer $n_i$ ($3 \le n_i \le 10^9$): it means that the $i$-th polygon is a regular $n_i$-sided polygon. 


-----Output-----

For each polygon, print YES if it's beautiful or NO otherwise (case insensitive).


-----Example-----
Input
4
3
4
12
1000000000

Output
NO
YES
YES
YES



-----Note-----

In the example, there are $4$ polygons in the market. It's easy to see that an equilateral triangle (a regular $3$-sided polygon) is not beautiful, a square (a regular $4$-sided polygon) is beautiful and a regular $12$-sided polygon (is shown below) is beautiful as well. [Image]
user: You have a fence consisting of $n$ vertical boards. The width of each board is $1$. The height of the $i$-th board is $a_i$. You think that the fence is great if there is no pair of adjacent boards having the same height. More formally, the fence is great if and only if for all indices from $2$ to $n$, the condition $a_{i-1} \neq a_i$ holds.

Unfortunately, it is possible that now your fence is not great. But you can change it! You can increase the length of the $i$-th board by $1$, but you have to pay $b_i$ rubles for it. The length of each board can be increased any number of times (possibly, zero).

Calculate the minimum number of rubles you have to spend to make the fence great again!

You have to answer $q$ independent queries.


-----Input-----

The first line contains one integer $q$ ($1 \le q \le 3 \cdot 10^5$) — the number of queries.

The first line of each query contains one integers $n$ ($1 \le n \le 3 \cdot 10^5$) — the number of boards in the fence.

The following $n$ lines of each query contain the descriptions of the boards. The $i$-th line contains two integers $a_i$ and $b_i$ ($1 \le a_i, b_i \le 10^9$) — the length of the $i$-th board and the price for increasing it by $1$, respectively.

It is guaranteed that sum of all $n$ over all queries not exceed $3 \cdot 10^5$.

It is guaranteed that answer to each query will not exceed $10^{18}$.


-----Output-----

For each query print one integer — the minimum number of rubles you have to spend to make the fence great.


-----Example-----
Input
3
3
2 4
2 1
3 5
3
2 3
2 10
2 6
4
1 7
3 3
2 6
1000000000 2

Output
2
9
0



-----Note-----

In the first query you have to increase the length of second board by $2$. So your total costs if $2 \cdot b_2 = 2$.

In the second query you have to increase the length of first board by $1$ and the length of third board by $1$. So your total costs if $1 \cdot b_1 + 1 \cdot b_3 = 9$.

In the third query the fence is great initially, so you don't need to spend rubles.