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 c++17.

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:
```cpp
<your code here>
```

Execution time limit: 1.5 seconds
Memory limit: 256.0 MB

# Problem
You have a rectangular board of size $$$n\times m$$$ ($$$n$$$ rows, $$$m$$$ columns). The $$$n$$$ rows are numbered from $$$1$$$ to $$$n$$$ from top to bottom, and the $$$m$$$ columns are numbered from $$$1$$$ to $$$m$$$ from left to right.

The cell at the intersection of row $$$i$$$ and column $$$j$$$ contains the number $$$i^j$$$ ($$$i$$$ raised to the power of $$$j$$$). For example, if $$$n=3$$$ and $$$m=3$$$ the board is as follows:

Find the number of distinct integers written on the board.

## Input Format
The only line contains two integers $$$n$$$ and $$$m$$$ ($$$1\le n,m\le 10^6$$$) — the number of rows and columns of the board.

## Output Format
Print one integer, the number of distinct integers on the board.

## Examples
```input
3 3
```
```output
7
```
-----
```input
2 4
```
```output
5
```
-----
```input
4 2
```
```output
6
```

## Note
The statement shows the board for the first test case. In this case there are $$$7$$$ distinct integers: $$$1$$$, $$$2$$$, $$$3$$$, $$$4$$$, $$$8$$$, $$$9$$$, and $$$27$$$.

In the second test case, the board is as follows:

There are $$$5$$$ distinct numbers: $$$1$$$, $$$2$$$, $$$4$$$, $$$8$$$ and $$$16$$$.

In the third test case, the board is as follows:

There are $$$6$$$ distinct numbers: $$$1$$$, $$$2$$$, $$$3$$$, $$$4$$$, $$$9$$$ and $$$16$$$.

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 c++17.

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:
```cpp
<your code here>
```

Execution time limit: 2.0 seconds
Memory limit: 256.0 MB

# Problem
There are $$$n$$$ players, numbered from $$$1$$$ to $$$n$$$ sitting around a round table. The $$$(i+1)$$$-th player sits to the right of the $$$i$$$-th player for $$$1 \le i < n$$$, and the $$$1$$$-st player sits to the right of the $$$n$$$-th player.

There are $$$n^2$$$ cards, each of which has an integer between $$$1$$$ and $$$n$$$ written on it. For each integer from $$$1$$$ to $$$n$$$, there are exactly $$$n$$$ cards having this number.

Initially, all these cards are distributed among all the players, in such a way that each of them has exactly $$$n$$$ cards. In one operation, each player chooses one of his cards and passes it to the player to his right. All these actions are performed simultaneously.

Player $$$i$$$ is called solid if all his cards have the integer $$$i$$$ written on them. Their objective is to reach a configuration in which everyone is solid. Find a way to do it using at most $$$(n^2-n)$$$ operations. You do not need to minimize the number of operations.

## Input Format
The first line contains a single integer $$$n$$$ ($$$2\le n\le 100$$$).

Then $$$n$$$ lines follow. The $$$i$$$-th of them contains $$$n$$$ integers $$$c_1, c_2, \ldots, c_n$$$ ($$$1\le c_j\le n$$$) — the initial cards of the $$$i$$$-th player.

It is guaranteed that for each integer $$$i$$$ from $$$1$$$ to $$$n$$$, there are exactly $$$n$$$ cards having the number $$$i$$$.

## Output Format
In the first line print an integer $$$k$$$ ($$$0\le k\le (n^2-n)$$$) — the numbers of operations you want to make.

Then $$$k$$$ lines should follow. In the $$$i$$$-th of them print $$$n$$$ integers $$$d_1, d_2, \ldots, d_n$$$ ($$$1\le d_j\le n$$$) where $$$d_j$$$ is the number written on the card which $$$j$$$-th player passes to the player to his right in the $$$i$$$-th operation.

We can show that an answer always exists under the given constraints. If there are multiple answers, print any.

## Examples
```input
2
2 1
1 2
```
```output
1
2 1
```
-----
```input
3
1 1 1
2 2 2
3 3 3
```
```output
6
1 2 3
3 1 2
2 3 1
1 2 3
3 1 2
2 3 1
```

## Note
In the first test case, if the first player passes a card with number $$$2$$$ and the second player passes a card with number $$$1$$$, then the first player has two cards with number $$$1$$$ and the second player has two cards with number $$$2$$$. Then, after making this operation, both players are solid.

In the second test case, $$$0$$$ operations would be enough too. Note that you do not need to minimize the number of operations.

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
The string $$$s$$$ is given, the string length is odd number. The string consists of lowercase letters of the Latin alphabet.

As long as the string length is greater than $$$1$$$, the following operation can be performed on it: select any two adjacent letters in the string $$$s$$$ and delete them from the string. For example, from the string "lemma" in one operation, you can get any of the four strings: "mma", "lma", "lea" or "lem" In particular, in one operation, the length of the string reduces by $$$2$$$.

Formally, let the string $$$s$$$ have the form $$$s=s_1s_2 \dots s_n$$$ ($$$n>1$$$). During one operation, you choose an arbitrary index $$$i$$$ ($$$1 \le i < n$$$) and replace $$$s=s_1s_2 \dots s_{i-1}s_{i+2} \dots s_n$$$.

For the given string $$$s$$$ and the letter $$$c$$$, determine whether it is possible to make such a sequence of operations that in the end the equality $$$s=c$$$ will be true? In other words, is there such a sequence of operations that the process will end with a string of length $$$1$$$, which consists of the letter $$$c$$$?

## Input Format
The first line of input data contains an integer $$$t$$$ ($$$1 \le t \le 10^3$$$) — the number of input test cases.

The descriptions of the $$$t$$$ cases follow. Each test case is represented by two lines:

- string $$$s$$$, which has an odd length from $$$1$$$ to $$$49$$$ inclusive and consists of lowercase letters of the Latin alphabet;
- is a string containing one letter $$$c$$$, where $$$c$$$ is a lowercase letter of the Latin alphabet.

## Output Format
For each test case in a separate line output:

- YES, if the string $$$s$$$ can be converted so that $$$s=c$$$ is true;
- NO otherwise.

You can output YES and NO in any case (for example, the strings yEs, yes, Yes and YES will be recognized as a positive response).

## Examples
```input
5
abcde
c
abcde
b
x
y
aaaaaaaaaaaaaaa
a
contest
t
```
```output
YES
NO
NO
YES
YES
```

## Note
In the first test case, $$$s$$$="abcde". You need to get $$$s$$$="c". For the first operation, delete the first two letters, we get $$$s$$$="cde". In the second operation, we delete the last two letters, so we get the expected value of $$$s$$$="c".

In the third test case, $$$s$$$="x", it is required to get $$$s$$$="y". Obviously, this cannot be done.

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
Not so long ago, Vlad came up with an interesting function:

- $$$f_a(x)=\left\lfloor\frac{x}{a}\right\rfloor + x \bmod a$$$, where $$$\left\lfloor\frac{x}{a}\right\rfloor$$$ is $$$\frac{x}{a}$$$, rounded down, $$$x \bmod a$$$ — the remainder of the integer division of $$$x$$$ by $$$a$$$.

For example, with $$$a=3$$$ and $$$x=11$$$, the value $$$f_3(11) = \left\lfloor\frac{11}{3}\right\rfloor + 11 \bmod 3 = 3 + 2 = 5$$$.

The number $$$a$$$ is fixed and known to Vlad. Help Vlad find the maximum value of $$$f_a(x)$$$ if $$$x$$$ can take any integer value from $$$l$$$ to $$$r$$$ inclusive ($$$l \le x \le r$$$).

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

This is followed by $$$t$$$ lines, each of which contains three integers $$$l_i$$$, $$$r_i$$$ and $$$a_i$$$ ($$$1 \le l_i \le r_i \le 10^9, 1 \le a_i \le 10^9$$$) — the left and right boundaries of the segment and the fixed value of $$$a$$$.

## Output Format
For each test case, output one number on a separate line — the maximum value of the function on a given segment for a given $$$a$$$.

## Examples
```input
5
1 4 3
5 8 4
6 10 6
1 1000000000 1000000000
10 12 8
```
```output
2
4
5
999999999
5
```

## Note
In the first sample:

- $$$f_3(1) = \left\lfloor\frac{1}{3}\right\rfloor + 1 \bmod 3 = 0 + 1 = 1$$$,
- $$$f_3(2) = \left\lfloor\frac{2}{3}\right\rfloor + 2 \bmod 3 = 0 + 2 = 2$$$,
- $$$f_3(3) = \left\lfloor\frac{3}{3}\right\rfloor + 3 \bmod 3 = 1 + 0 = 1$$$,
- $$$f_3(4) = \left\lfloor\frac{4}{3}\right\rfloor + 4 \bmod 3 = 1 + 1 = 2$$$

As an answer, obviously, $$$f_3(2)$$$ and $$$f_3(4)$$$ are suitable.

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 arrays of length $$$n$$$: $$$a_1, a_2, \dots, a_n$$$ and $$$b_1, b_2, \dots, b_n$$$.

You can perform the following operation any number of times:

1. Choose integer index $$$i$$$ ($$$1 \le i \le n$$$);
2. Swap $$$a_i$$$ and $$$b_i$$$.

What is the minimum possible sum $$$|a_1 - a_2| + |a_2 - a_3| + \dots + |a_{n-1} - a_n|$$$ $$$+$$$ $$$|b_1 - b_2| + |b_2 - b_3| + \dots + |b_{n-1} - b_n|$$$ (in other words, $$$\sum\limits_{i=1}^{n - 1}{\left(|a_i - a_{i+1}| + |b_i - b_{i+1}|\right)}$$$) you can achieve after performing several (possibly, zero) operations?

## Input Format
The first line contains a single integer $$$t$$$ ($$$1 \le t \le 4000$$$) — the number of test cases. Then, $$$t$$$ test cases follow.

The first line of each test case contains the single integer $$$n$$$ ($$$2 \le n \le 25$$$) — the length of arrays $$$a$$$ and $$$b$$$.

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

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

## Output Format
For each test case, print one integer — the minimum possible sum $$$\sum\limits_{i=1}^{n-1}{\left(|a_i - a_{i+1}| + |b_i - b_{i+1}|\right)}$$$.

## Examples
```input
3
4
3 3 10 10
10 10 3 3
5
1 2 3 4 5
6 7 8 9 10
6
72 101 108 108 111 44
10 87 111 114 108 100
```
```output
0
8
218
```

## Note
In the first test case, we can, for example, swap $$$a_3$$$ with $$$b_3$$$ and $$$a_4$$$ with $$$b_4$$$. We'll get arrays $$$a = [3, 3, 3, 3]$$$ and $$$b = [10, 10, 10, 10]$$$ with sum $$$3 \cdot |3 - 3| + 3 \cdot |10 - 10| = 0$$$.

In the second test case, arrays already have minimum sum (described above) equal to $$$|1 - 2| + \dots + |4 - 5| + |6 - 7| + \dots + |9 - 10|$$$ $$$= 4 + 4 = 8$$$.

In the third test case, we can, for example, swap $$$a_5$$$ and $$$b_5$$$.

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
Suppose you have an integer $$$v$$$. In one operation, you can:

- either set $$$v = (v + 1) \bmod 32768$$$
- or set $$$v = (2 \cdot v) \bmod 32768$$$.

You are given $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$. What is the minimum number of operations you need to make each $$$a_i$$$ equal to $$$0$$$?

## Input Format
The first line contains the single integer $$$n$$$ ($$$1 \le n \le 32768$$$) — the number of integers.

The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$0 \le a_i < 32768$$$).

## Output Format
Print $$$n$$$ integers. The $$$i$$$-th integer should be equal to the minimum number of operations required to make $$$a_i$$$ equal to $$$0$$$.

## Examples
```input
4
19 32764 10240 49
```
```output
14 4 4 15
```

## Note
Let's consider each $$$a_i$$$:

- $$$a_1 = 19$$$. You can, firstly, increase it by one to get $$$20$$$ and then multiply it by two $$$13$$$ times. You'll get $$$0$$$ in $$$1 + 13 = 14$$$ steps.
- $$$a_2 = 32764$$$. You can increase it by one $$$4$$$ times: $$$32764 \rightarrow 32765 \rightarrow 32766 \rightarrow 32767 \rightarrow 0$$$.
- $$$a_3 = 10240$$$. You can multiply it by two $$$4$$$ times: $$$10240 \rightarrow 20480 \rightarrow 8192 \rightarrow 16384 \rightarrow 0$$$.
- $$$a_4 = 49$$$. You can multiply it by two $$$15$$$ times.

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: 3.0 seconds
Memory limit: 256.0 MB

# Problem
There are $$$n$$$ trees in a park, numbered from $$$1$$$ to $$$n$$$. The initial height of the $$$i$$$-th tree is $$$h_i$$$.

You want to water these trees, so they all grow to the same height.

The watering process goes as follows. You start watering trees at day $$$1$$$. During the $$$j$$$-th day you can:

- Choose a tree and water it. If the day is odd (e.g. $$$1, 3, 5, 7, \dots$$$), then the height of the tree increases by $$$1$$$. If the day is even (e.g. $$$2, 4, 6, 8, \dots$$$), then the height of the tree increases by $$$2$$$.
- Or skip a day without watering any tree.

Note that you can't water more than one tree in a day.

Your task is to determine the minimum number of days required to water the trees so they grow to the same height.

You have to answer $$$t$$$ independent test cases.

## Input Format
The first line of the input contains one integer $$$t$$$ ($$$1 \le t \le 2 \cdot 10^4$$$) — the number of test cases.

The first line of the test case contains one integer $$$n$$$ ($$$1 \le n \le 3 \cdot 10^5$$$) — the number of trees.

The second line of the test case contains $$$n$$$ integers $$$h_1, h_2, \ldots, h_n$$$ ($$$1 \le h_i \le 10^9$$$), where $$$h_i$$$ is the height of the $$$i$$$-th tree.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$3 \cdot 10^5$$$ ($$$\sum n \le 3 \cdot 10^5$$$).

## Output Format
For each test case, print one integer — the minimum number of days required to water the trees, so they grow to the same height.

## Examples
```input
3
3
1 2 4
5
4 4 3 5 5
7
2 5 4 8 3 7 4
```
```output
4
3
16
```

## Note
Consider the first test case of the example. The initial state of the trees is $$$[1, 2, 4]$$$.

1. During the first day, let's water the first tree, so the sequence of heights becomes $$$[2, 2, 4]$$$;
2. during the second day, let's water the second tree, so the sequence of heights becomes $$$[2, 4, 4]$$$;
3. let's skip the third day;
4. during the fourth day, let's water the first tree, so the sequence of heights becomes $$$[4, 4, 4]$$$.

Thus, the answer is $$$4$$$.

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 arrays: an array $$$a$$$ consisting of $$$n$$$ zeros and an array $$$b$$$ consisting of $$$n$$$ integers.

You can apply the following operation to the array $$$a$$$ an arbitrary number of times: choose some subsegment of $$$a$$$ of length $$$k$$$ and add the arithmetic progression $$$1, 2, \ldots, k$$$ to this subsegment — i. e. add $$$1$$$ to the first element of the subsegment, $$$2$$$ to the second element, and so on. The chosen subsegment should be inside the borders of the array $$$a$$$ (i.e., if the left border of the chosen subsegment is $$$l$$$, then the condition $$$1 \le l \le l + k - 1 \le n$$$ should be satisfied). Note that the progression added is always $$$1, 2, \ldots, k$$$ but not the $$$k, k - 1, \ldots, 1$$$ or anything else (i.e., the leftmost element of the subsegment always increases by $$$1$$$, the second element always increases by $$$2$$$ and so on).

Your task is to find the minimum possible number of operations required to satisfy the condition $$$a_i \ge b_i$$$ for each $$$i$$$ from $$$1$$$ to $$$n$$$. Note that the condition $$$a_i \ge b_i$$$ should be satisfied for all elements at once.

## Input Format
The first line of the input contains two integers $$$n$$$ and $$$k$$$ ($$$1 \le k \le n \le 3 \cdot 10^5$$$) — the number of elements in both arrays and the length of the subsegment, respectively.

The second line of the input contains $$$n$$$ integers $$$b_1, b_2, \ldots, b_n$$$ ($$$1 \le b_i \le 10^{12}$$$), where $$$b_i$$$ is the $$$i$$$-th element of the array $$$b$$$.

## Output Format
Print one integer — the minimum possible number of operations required to satisfy the condition $$$a_i \ge b_i$$$ for each $$$i$$$ from $$$1$$$ to $$$n$$$.

## Examples
```input
3 3
5 4 6
```
```output
5
```
-----
```input
6 3
1 2 3 2 2 3
```
```output
3
```
-----
```input
6 3
1 2 4 1 2 3
```
```output
3
```
-----
```input
7 3
50 17 81 25 42 39 96
```
```output
92
```

## Note
Consider the first example. In this test, we don't really have any choice, so we need to add at least five progressions to make the first element equals $$$5$$$. The array $$$a$$$ becomes $$$[5, 10, 15]$$$.

Consider the second example. In this test, let's add one progression on the segment $$$[1; 3]$$$ and two progressions on the segment $$$[4; 6]$$$. Then, the array $$$a$$$ becomes $$$[1, 2, 3, 2, 4, 6]$$$.

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: 7.0 seconds
Memory limit: 512.0 MB

# Problem
There are $$$n+1$$$ teleporters on a straight line, located in points $$$0$$$, $$$a_1$$$, $$$a_2$$$, $$$a_3$$$, ..., $$$a_n$$$. It's possible to teleport from point $$$x$$$ to point $$$y$$$ if there are teleporters in both of those points, and it costs $$$(x-y)^2$$$ energy.

You want to install some additional teleporters so that it is possible to get from the point $$$0$$$ to the point $$$a_n$$$ (possibly through some other teleporters) spending no more than $$$m$$$ energy in total. Each teleporter you install must be located in an integer point.

What is the minimum number of teleporters you have to install?

## Input Format
The first line contains one integer $$$n$$$ ($$$1 \le n \le 2 \cdot 10^5$$$).

The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_1 < a_2 < a_3 < \dots < a_n \le 10^9$$$).

The third line contains one integer $$$m$$$ ($$$a_n \le m \le 10^{18}$$$).

## Output Format
Print one integer — the minimum number of teleporters you have to install so that it is possible to get from $$$0$$$ to $$$a_n$$$ spending at most $$$m$$$ energy. It can be shown that it's always possible under the constraints from the input format.

## Examples
```input
2
1 5
7
```
```output
2
```
-----
```input
2
1 5
6
```
```output
3
```
-----
```input
1
5
5
```
```output
4
```
-----
```input
1
1000000000
1000000043
```
```output
999999978
```

Now solve the problem and return the code.