apaz/codeforces-genesys huggingface dataset:
You are an expert competitive programmer. You will be given a problem statement, test case constraints and example test inputs and outputs. Please reason step by step about the solution (that must respect memory and time limits), then provide a complete implementation in python3.

Your solution must read input from standard input (cin), write output to standard output (cout).
Do not include any debug prints or additional output.

Put your final solution within a single code block:
```python
<your code here>
```

Execution time limit: 1.0 seconds
Memory limit: 256.0 MB

# Problem
A sequence of $$$m$$$ integers $$$a_{1}, a_{2}, \ldots, a_{m}$$$ is good, if and only if there exists at least one $$$i$$$ ($$$1 \le i \le m$$$) such that $$$a_{i} = i$$$. For example, $$$[3,2,3]$$$ is a good sequence, since $$$a_{2} = 2$$$, $$$a_{3} = 3$$$, while $$$[3,1,1]$$$ is not a good sequence, since there is no $$$i$$$ such that $$$a_{i} = i$$$.

A sequence $$$a$$$ is beautiful, if and only if there exists at least one subsequence of $$$a$$$ satisfying that this subsequence is good. For example, $$$[4,3,2]$$$ is a beautiful sequence, since its subsequence $$$[4,2]$$$ is good, while $$$[5,3,4]$$$ is not a beautiful sequence.

A sequence $$$b$$$ is a subsequence of a sequence $$$a$$$ if $$$b$$$ can be obtained from $$$a$$$ by the deletion of several (possibly, zero or all) elements.

Now you are given a sequence, check whether it is beautiful or not.

## Input Format
Each test contains multiple test cases. The first line contains a single integer $$$t$$$ ($$$1 \le t \le 500$$$) — the number of test cases. Their description follows.

The first line of each test case contains an integer $$$n$$$ ($$$1 \le n \le 100$$$) — the length of the given sequence.

The second line of each test case contains $$$n$$$ integers $$$a_{1}, a_{2}, \ldots, a_{n}$$$ ($$$1 \le a_{i} \le 10^9$$$), representing the sequence.

## Output Format
For each test case, output "YES" or "NO"(without quotes) in one line, representing whether the given sequence is beautiful.

You can output the answer in any case (upper or lower). For example, the strings "yEs", "yes", "Yes", and "YES" will be recognized as positive responses.

## Examples
```input
7
3
3 2 1
4
2 4 3 5
5
2 3 5 5 6
2
3 1
5
2 4 5 2 3
4
5 6 7 8
6
6 5 4 3 2 1
```
```output
YES
YES
NO
YES
YES
NO
YES
```

## Note
In the first test case, the good subsequence is $$$b=[3,2]$$$, where $$$b_{2} = 2$$$.

In the second test case, the good subsequence is $$$b=[2,4,3]$$$, where $$$b_{3} = 3$$$.

In the fourth test case, the good subsequence is $$$b=[1]$$$, where $$$b_{1} = 1$$$.

In the fifth test case, the good subsequence is $$$b=[2,2]$$$, where $$$b_{2} = 2$$$.

Now solve the problem and return the code.
You are an expert competitive programmer. You will be given a problem statement, test case constraints and example test inputs and outputs. Please reason step by step about the solution (that must respect memory and time limits), then provide a complete implementation in python3.

Your solution must read input from standard input (cin), write output to standard output (cout).
Do not include any debug prints or additional output.

Put your final solution within a single code block:
```python
<your code here>
```

Execution time limit: 1.0 seconds
Memory limit: 256.0 MB

# Problem
This problem is about candy. Initially, you only have $$$1$$$ candy, and you want to have exactly $$$n$$$ candies.

You can use the two following spells in any order at most $$$40$$$ times in total.

- Assume you have $$$x$$$ candies now. If you use the first spell, then $$$x$$$ candies become $$$2x-1$$$ candies.
- Assume you have $$$x$$$ candies now. If you use the second spell, then $$$x$$$ candies become $$$2x+1$$$ candies.

Construct a sequence of spells, such that after using them in order, you will have exactly $$$n$$$ candies, or determine it's impossible.

## Input Format
Each test contains multiple test cases. The first line contains a single integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases. Their description follows.

Each test case contains one line with a single integer $$$n$$$ ($$$2 \le n \le 10^9$$$) — the required final number of candies.

## Output Format
For each test case, output the following.

If it's possible to eventually have $$$n$$$ candies within $$$40$$$ spells, in the first line print an integer $$$m$$$ ($$$1 \le m \le 40$$$), representing the total number of spells you use.

In the second print $$$m$$$ integers $$$a_{1}, a_{2}, \ldots, a_{m}$$$ ($$$a_{i}$$$ is $$$1$$$ or $$$2$$$) separated by spaces, where $$$a_{i} = 1$$$ means that you use the first spell in the $$$i$$$-th step, while $$$a_{i} = 2$$$ means that you use the second spell in the $$$i$$$-th step.

Note that you do not have to minimize $$$m$$$, and if there are multiple solutions, you may output any one of them.

If it's impossible, output $$$-1$$$ in one line.

## Examples
```input
4
2
3
7
17
```
```output
-1
1
2 
2
2 2 
4
2 1 1 1
```

## Note
For $$$n=3$$$, you can just use the second spell once, and then have $$$2 \cdot 1 + 1 = 3$$$ candies.

For $$$n=7$$$, you can use the second spell twice. After the first step, you will have $$$3$$$ candies. And after the second step, you will have $$$2 \cdot 3 + 1 = 7$$$ candies.

Now solve the problem and return the code.
You are an expert competitive programmer. You will be given a problem statement, test case constraints and example test inputs and outputs. Please reason step by step about the solution (that must respect memory and time limits), then provide a complete implementation in python3.

Your solution must read input from standard input (cin), write output to standard output (cout).
Do not include any debug prints or additional output.

Put your final solution within a single code block:
```python
<your code here>
```

Execution time limit: 1.0 seconds
Memory limit: 256.0 MB

# Problem
You have an integer array $$$a$$$ of length $$$n$$$. There are two kinds of operations you can make.

- Remove an integer from $$$a$$$. This operation costs $$$c$$$.
- Insert an arbitrary positive integer $$$x$$$ to any position of $$$a$$$ (to the front, to the back, or between any two consecutive elements). This operation costs $$$d$$$.

You want to make the final array a permutation of any positive length. Please output the minimum cost of doing that. Note that you can make the array empty during the operations, but the final array must contain at least one integer.

A permutation of length $$$n$$$ is an array consisting of $$$n$$$ distinct integers from $$$1$$$ to $$$n$$$ in arbitrary order. For example, $$$[2,3,1,5,4]$$$ is a permutation, but $$$[1,2,2]$$$ is not a permutation ($$$2$$$ appears twice in the array), and $$$[1,3,4]$$$ is also not a permutation ($$$n=3$$$ but there is $$$4$$$ in the array).

## Input Format
Each test contains multiple test cases. The first line contains a single integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases. Their description follows.

The first line of each test case contains three integers $$$n$$$, $$$c$$$, $$$d$$$ ($$$1 \le n \le 10^5$$$, $$$1 \le c,d \le 10^9$$$).

The second line of each test case contains $$$n$$$ integers $$$a_{1}, a_{2}, \ldots, a_{n}$$$ ($$$1 \le a_{i} \le 10^9$$$).

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$.

## Output Format
For each test case, output in one line the minimum cost to make the final array a permutation.

## Examples
```input
8
3 3 3
1 2 3
5 1 5
1 2 3 5 6
5 2 3
1 1 1 3 3
5 1 10
2 4 6 8 10
6 2 8
7 3 5 4 4 8
4 10 1
1 2 6 7
4 3 3
2 5 8 7
2 1000000000 1
1000000000 1
```
```output
0
2
8
14
20
3
12
999999998
```

## Note
In the first test case, the array is already a permutation, so there's no need for operations.

In the second test case, we can remove numbers $$$5$$$, $$$6$$$ to get the permutation $$$[1,2,3]$$$ in cost $$$2$$$. Note that we can also get a permutation by inserting a number $$$4$$$, but it costs $$$5$$$.

In the third test case, we can just remove all the numbers except for the first number $$$1$$$. It costs $$$8$$$, and the final array is $$$[1]$$$, which is a permutation of length $$$1$$$.

In the fourth test case, we can remove all the numbers except for $$$2$$$, and insert a number $$$1$$$ to the first position. It costs $$$4+10=14$$$, and the final array is $$$[1,2]$$$, which is a permutation of length $$$2$$$.

Now solve the problem and return the code.
You are an expert competitive programmer. You will be given a problem statement, test case constraints and example test inputs and outputs. Please reason step by step about the solution (that must respect memory and time limits), then provide a complete implementation in python3.

Your solution must read input from standard input (cin), write output to standard output (cout).
Do not include any debug prints or additional output.

Put your final solution within a single code block:
```python
<your code here>
```

Execution time limit: 2.0 seconds
Memory limit: 256.0 MB

# Problem
You are given a multiset $$$S$$$. Initially, $$$S = \{1,2,3, \ldots, n\}$$$.

You will perform the following operation $$$n-1$$$ times.

- Choose the largest number $$$S_{\text{max}}$$$ in $$$S$$$ and the smallest number $$$S_{\text{min}}$$$ in $$$S$$$. Remove the two numbers from $$$S$$$, and add $$$S_{\text{max}} - S_{\text{min}}$$$ into $$$S$$$.

It's easy to show that there will be exactly one number left after $$$n-1$$$ operations. Output that number.

## Input Format
Each test contains multiple test cases. The first line contains a single integer $$$t$$$ ($$$1 \le t \le 10^5$$$) — the number of test cases. Their description follows.

For each test case, one single line contains a single integer $$$n$$$ ($$$2 \le n \le 10^9$$$) — the initial size of the multiset $$$S$$$.

## Output Format
For each test case, output an integer denoting the only number left after $$$n-1$$$ operations.

## Examples
```input
5
2
4
7
15
177567
```
```output
1
2
2
4
33914
```

## Note
We show how the multiset $$$S$$$ changes for $$$n=4$$$.

- Operation $$$1$$$: $$$S=\{1,2,3,4\}$$$, remove $$$4$$$, $$$1$$$, add $$$3$$$.
- Operation $$$2$$$: $$$S=\{2,3,3\}$$$, remove $$$3$$$, $$$2$$$, add $$$1$$$.
- Operation $$$3$$$: $$$S=\{1,3\}$$$, remove $$$3$$$, $$$1$$$, add $$$2$$$.
- Final: $$$S = \{2\}$$$.

Thus, the answer for $$$n = 4$$$ is $$$2$$$.

Now solve the problem and return the code.
You are an expert competitive programmer. You will be given a problem statement, test case constraints and example test inputs and outputs. Please reason step by step about the solution (that must respect memory and time limits), then provide a complete implementation in python3.

Your solution must read input from standard input (cin), write output to standard output (cout).
Do not include any debug prints or additional output.

Put your final solution within a single code block:
```python
<your code here>
```

Execution time limit: 2.0 seconds
Memory limit: 256.0 MB

# Problem
You are given two integers $$$x$$$ and $$$k$$$. Grasshopper starts in a point $$$0$$$ on an OX axis. In one move, it can jump some integer distance, that is not divisible by $$$k$$$, to the left or to the right.

What's the smallest number of moves it takes the grasshopper to reach point $$$x$$$? What are these moves? If there are multiple answers, print any of them.

## Input Format
The first line contains a single integer $$$t$$$ ($$$1 \le t \le 1000$$$) — the number of testcases.

The only line of each testcase contains two integers $$$x$$$ and $$$k$$$ ($$$1 \le x \le 100$$$; $$$2 \le k \le 100$$$) — the endpoint and the constraint on the jumps, respectively.

## Output Format
For each testcase, in the first line, print a single integer $$$n$$$ — the smallest number of moves it takes the grasshopper to reach point $$$x$$$.

In the second line, print $$$n$$$ integers, each of them not divisible by $$$k$$$. A positive integer would mean jumping to the right, a negative integer would mean jumping to the left. The endpoint after the jumps should be exactly $$$x$$$.

Each jump distance should be from $$$-10^9$$$ to $$$10^9$$$. In can be shown that, for any solution with the smallest number of jumps, there exists a solution with the same number of jumps such that each jump is from $$$-10^9$$$ to $$$10^9$$$.

It can be shown that the answer always exists under the given constraints. If there are multiple answers, print any of them.

## Examples
```input
3
10 2
10 3
3 4
```
```output
2
7 3
1
10
1
3
```

Now solve the problem and return the code.
You are an expert competitive programmer. You will be given a problem statement, test case constraints and example test inputs and outputs. Please reason step by step about the solution (that must respect memory and time limits), then provide a complete implementation in python3.

Your solution must read input from standard input (cin), write output to standard output (cout).
Do not include any debug prints or additional output.

Put your final solution within a single code block:
```python
<your code here>
```

Execution time limit: 2.0 seconds
Memory limit: 512.0 MB

# Problem
You are given a string $$$s$$$ of length $$$n$$$, where each character is either < or >.

An array $$$a$$$ consisting of $$$n+1$$$ elements is compatible with the string $$$s$$$ if, for every $$$i$$$ from $$$1$$$ to $$$n$$$, the character $$$s_i$$$ represents the result of comparing $$$a_i$$$ and $$$a_{i+1}$$$, i. e.:

- $$$s_i$$$ is < if and only if $$$a_i < a_{i+1}$$$;
- $$$s_i$$$ is > if and only if $$$a_i > a_{i+1}$$$.

For example, the array $$$[1, 2, 5, 4, 2]$$$ is compatible with the string <<>>. There are other arrays with are compatible with that string, for example, $$$[13, 37, 42, 37, 13]$$$.

The cost of the array is the number of different elements in it. For example, the cost of $$$[1, 2, 5, 4, 2]$$$ is $$$4$$$; the cost of $$$[13, 37, 42, 37, 13]$$$ is $$$3$$$.

You have to calculate the minimum cost among all arrays which are compatible with the given string $$$s$$$.

## Input Format
The first line contains one integer $$$t$$$ ($$$1 \le t \le 500$$$) — the number of test cases.

Each test case consists of two lines:

- the first line contains one integer $$$n$$$ ($$$1 \le n \le 100$$$);
- the second line contains the string $$$s$$$, consisting of $$$n$$$ characters. Each character of $$$s$$$ is either < or >.

## Output Format
For each test case, print one integer — the minimum cost among all arrays which are compatible with the given string $$$s$$$.

## Examples
```input
4
4
<<>>
4
>><<
5
>>>>>
7
<><><><
```
```output
3
3
6
2
```

## Note
In the first test case of the example, the array can be $$$[13, 37, 42, 37, 13]$$$.

In the second test case of the example, the array can be $$$[42, 37, 13, 37, 42]$$$.

Now solve the problem and return the code.
You are an expert competitive programmer. You will be given a problem statement, test case constraints and example test inputs and outputs. Please reason step by step about the solution (that must respect memory and time limits), then provide a complete implementation in python3.

Your solution must read input from standard input (cin), write output to standard output (cout).
Do not include any debug prints or additional output.

Put your final solution within a single code block:
```python
<your code here>
```

Execution time limit: 4.0 seconds
Memory limit: 256.0 MB

# Problem
Berland Intercollegiate Contest has just finished. Monocarp and Polycarp, as the jury, are going to conduct an editorial. Unfortunately, the time is limited, since they have to finish before the closing ceremony.

There were $$$n$$$ problems in the contest. The problems are numbered from $$$1$$$ to $$$n$$$. The editorial for the $$$i$$$-th problem takes $$$a_i$$$ minutes. Monocarp and Polycarp are going to conduct an editorial for exactly $$$k$$$ of the problems.

The editorial goes as follows. They have a full problemset of $$$n$$$ problems before them, in order. They remove $$$n - k$$$ problems without changing the order of the remaining $$$k$$$ problems. Then, Monocarp takes some prefix of these $$$k$$$ problems (possibly, an empty one or all problems). Polycarp takes the remaining suffix of them. After that, they go to different rooms and conduct editorials for their problems in parallel. So, the editorial takes as much time as the longer of these two does.

Please, help Monocarp and Polycarp to choose the problems and the split in such a way that the editorial finishes as early as possible. Print the duration of the editorial.

## Input Format
The first line contains a single integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of testcases.

The first line of each testcase contains two integers $$$n$$$ and $$$k$$$ ($$$1 \le k \le n \le 3 \cdot 10^5$$$) — the number of problems in the full problemset and the number of problems Monocarp and Polycarp are going to conduct an editorial for.

The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 10^9$$$) — the time each editorial takes.

The sum of $$$n$$$ over all testcases doesn't exceed $$$3 \cdot 10^5$$$.

## Output Format
For each testcase, print a single integer — the smallest amount of time the editorial takes, if Monocarp and Polycarp can choose which $$$k$$$ of $$$n$$$ problems to conduct an editorial for and how to split them among themselves.

## Examples
```input
6
5 4
1 10 1 1 1
5 3
1 20 5 15 3
5 3
1 20 3 15 5
10 6
10 8 20 14 3 8 6 4 16 11
10 5
9 9 2 13 15 19 4 9 13 12
1 1
1
```
```output
2
6
5
21
18
1
```

Now solve the problem and return the code.
You are an expert competitive programmer. You will be given a problem statement, test case constraints and example test inputs and outputs. Please reason step by step about the solution (that must respect memory and time limits), then provide a complete implementation in python3.

Your solution must read input from standard input (cin), write output to standard output (cout).
Do not include any debug prints or additional output.

Put your final solution within a single code block:
```python
<your code here>
```

Execution time limit: 1.0 seconds
Memory limit: 256.0 MB

# Problem
Dalton is the teacher of a class with $$$n$$$ students, numbered from $$$1$$$ to $$$n$$$. The classroom contains $$$n$$$ chairs, also numbered from $$$1$$$ to $$$n$$$. Initially student $$$i$$$ is seated on chair $$$p_i$$$. It is guaranteed that $$$p_1,p_2,\dots, p_n$$$ is a permutation of length $$$n$$$.

A student is happy if his/her number is different from the number of his/her chair. In order to make all of his students happy, Dalton can repeatedly perform the following operation: choose two distinct students and swap their chairs. What is the minimum number of moves required to make all the students happy? One can show that, under the constraints of this problem, it is possible to make all the students happy with a finite number of moves.

A permutation of length $$$n$$$ is an array consisting of $$$n$$$ distinct integers from $$$1$$$ to $$$n$$$ in arbitrary order. For example, $$$[2,3,1,5,4]$$$ is a permutation, but $$$[1,2,2]$$$ is not a permutation ($$$2$$$ appears twice in the array), and $$$[1,3,4]$$$ is also not a permutation ($$$n=3$$$ but there is $$$4$$$ in the array).

## Input Format
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 1000$$$). The description of the test cases follows.

The first line contains a single integer $$$n$$$ ($$$2 \le n \le 10^5$$$) — the number of students.

The second line contains $$$n$$$ integers $$$p_1, p_2, \ldots, p_n$$$ ($$$1 \le p_i \le n$$$) — $$$p_i$$$ denotes the initial chair of student $$$i$$$. It is guaranteed that $$$p$$$ is a permutation.

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

## Output Format
For each test case, output the minimum number of moves required.

## Examples
```input
5
2
2 1
3
1 2 3
5
1 2 5 4 3
4
1 2 4 3
10
10 2 1 3 6 5 4 7 9 8
```
```output
0
2
2
1
1
```

## Note
In the first test case, both students are already happy, so Dalton can perform $$$0$$$ moves.

In the second test case, Dalton can swap the chairs of students $$$1$$$ and $$$2$$$ to get the array $$$[2, 1, 3]$$$. Then he can swap chairs of students $$$2$$$ and $$$3$$$ to get the array $$$[2, 3, 1]$$$. At this point all the students are happy, and he performed $$$2$$$ moves. It is impossible to perform the task with fewer moves.

In the third test case, by swapping the chairs of students $$$1$$$ and $$$2$$$ and then swapping the chairs of students $$$4$$$ and $$$5$$$, Dalton gets the array $$$[2, 1, 5, 3, 4]$$$ in $$$2$$$ moves.

Now solve the problem and return the code.
You are an expert competitive programmer. You will be given a problem statement, test case constraints and example test inputs and outputs. Please reason step by step about the solution (that must respect memory and time limits), then provide a complete implementation in python3.

Your solution must read input from standard input (cin), write output to standard output (cout).
Do not include any debug prints or additional output.

Put your final solution within a single code block:
```python
<your code here>
```

Execution time limit: 2.0 seconds
Memory limit: 256.0 MB

# Problem
Given a positive integer $$$n$$$, find the maximum size of an interval $$$[l, r]$$$ of positive integers such that, for every $$$i$$$ in the interval (i.e., $$$l \leq i \leq r$$$), $$$n$$$ is a multiple of $$$i$$$.

Given two integers $$$l\le r$$$, the size of the interval $$$[l, r]$$$ is $$$r-l+1$$$ (i.e., it coincides with the number of integers belonging to the interval).

## Input Format
The first line contains a single integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases.

The only line of the description of each test case contains one integer $$$n$$$ ($$$1 \leq n \leq 10^{18}$$$).

## Output Format
For each test case, print a single integer: the maximum size of a valid interval.

## Examples
```input
10
1
40
990990
4204474560
169958913706572972
365988220345828080
387701719537826430
620196883578129853
864802341280805662
1000000000000000000
```
```output
1
2
3
6
4
22
3
1
2
2
```

## Note
In the first test case, a valid interval with maximum size is $$$[1, 1]$$$ (it's valid because $$$n = 1$$$ is a multiple of $$$1$$$) and its size is $$$1$$$.

In the second test case, a valid interval with maximum size is $$$[4, 5]$$$ (it's valid because $$$n = 40$$$ is a multiple of $$$4$$$ and $$$5$$$) and its size is $$$2$$$.

In the third test case, a valid interval with maximum size is $$$[9, 11]$$$.

In the fourth test case, a valid interval with maximum size is $$$[8, 13]$$$.

In the seventh test case, a valid interval with maximum size is $$$[327869, 327871]$$$.

Now solve the problem and return the code.