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: 2.0 seconds
Memory limit: 512.0 MB

# Problem
Initially, we had one array, which was a permutation of size $$$n$$$ (an array of size $$$n$$$ where each integer from $$$1$$$ to $$$n$$$ appears exactly once).

We performed $$$q$$$ operations. During the $$$i$$$-th operation, we did the following:

- choose any array we have with at least $$$2$$$ elements;
- split it into two non-empty arrays (prefix and suffix);
- write two integers $$$l_i$$$ and $$$r_i$$$, where $$$l_i$$$ is the maximum element in the left part which we get after the split, and $$$r_i$$$ is the maximum element in the right part;
- remove the array we've chosen from the pool of arrays we can use, and add the two resulting parts into the pool.

For example, suppose the initial array was $$$[6, 3, 4, 1, 2, 5]$$$, and we performed the following operations:

1. choose the array $$$[6, 3, 4, 1, 2, 5]$$$ and split it into $$$[6, 3]$$$ and $$$[4, 1, 2, 5]$$$. Then we write $$$l_1 = 6$$$ and $$$r_1 = 5$$$, and the arrays we have are $$$[6, 3]$$$ and $$$[4, 1, 2, 5]$$$;
2. choose the array $$$[4, 1, 2, 5]$$$ and split it into $$$[4, 1, 2]$$$ and $$$[5]$$$. Then we write $$$l_2 = 4$$$ and $$$r_2 = 5$$$, and the arrays we have are $$$[6, 3]$$$, $$$[4, 1, 2]$$$ and $$$[5]$$$;
3. choose the array $$$[4, 1, 2]$$$ and split it into $$$[4]$$$ and $$$[1, 2]$$$. Then we write $$$l_3 = 4$$$ and $$$r_3 = 2$$$, and the arrays we have are $$$[6, 3]$$$, $$$[4]$$$, $$$[1, 2]$$$ and $$$[5]$$$.

You are given two integers $$$n$$$ and $$$q$$$, and two sequences $$$[l_1, l_2, \dots, l_q]$$$ and $$$[r_1, r_2, \dots, r_q]$$$. A permutation of size $$$n$$$ is called valid if we can perform $$$q$$$ operations and produce the given sequences $$$[l_1, l_2, \dots, l_q]$$$ and $$$[r_1, r_2, \dots, r_q]$$$.

Calculate the number of valid permutations.

## Input Format
The first line contains two integers $$$n$$$ and $$$q$$$ ($$$1 \le q < n \le 3 \cdot 10^5$$$).

The second line contains $$$q$$$ integers $$$l_1, l_2, \dots, l_q$$$ ($$$1 \le l_i \le n$$$).

The third line contains $$$q$$$ integers $$$r_1, r_2, \dots, r_q$$$ ($$$1 \le r_i \le n$$$).

Additional constraint on the input: there exists at least one permutation which can produce the given sequences $$$[l_1, l_2, \dots, l_q]$$$ and $$$[r_1, r_2, \dots, r_q]$$$.

## Output Format
Print one integer — the number of valid permutations, taken modulo $$$998244353$$$.

## Examples
```input
6 3
6 4 4
5 5 2
```
```output
30
```
-----
```input
10 1
10
9
```
```output
1814400
```
-----
```input
4 1
2
4
```
```output
8
```

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
Recently, the first-year student Maxim learned about the Collatz conjecture, but he didn't pay much attention during the lecture, so he believes that the following process is mentioned in the conjecture:

There is a variable $$$x$$$ and a constant $$$y$$$. The following operation is performed $$$k$$$ times:

- increase $$$x$$$ by $$$1$$$, then
- while the number $$$x$$$ is divisible by $$$y$$$, divide it by $$$y$$$.

For example, if the number $$$x = 16$$$, $$$y = 3$$$, and $$$k = 2$$$, then after one operation $$$x$$$ becomes $$$17$$$, and after another operation $$$x$$$ becomes $$$2$$$, because after adding one, $$$x = 18$$$ is divisible by $$$3$$$ twice.

Given the initial values of $$$x$$$, $$$y$$$, and $$$k$$$, Maxim wants to know what is the final value of $$$x$$$.

## Input Format
Each test consists of multiple test cases. The first line contains an integer $$$t$$$ ($$$1 \le t \le 10^{4}$$$) — the number of test cases. Then follows the description of the test cases.

The only line of each test case contains three integers $$$x$$$, $$$y$$$, and $$$k$$$ ($$$1 \le x, k \le 10^{9}$$$, $$$2 \le y \le 10^{9}$$$) — the initial variable, constant and the number of operations.

## Output Format
For each test case, output a single integer — the number obtained after applying $$$k$$$ operations.

## Examples
```input
13
1 3 1
2 3 1
24 5 5
16 3 2
2 2 1
1337 18 1
1 2 144133
12345678 3 10
998244353 2 998244353
998244353 123456789 998244352
998244354 998241111 998244352
998244355 2 9982443
1000000000 1000000000 1000000000
```
```output
2
1
1
2
3
1338
1
16936
1
21180097
6486
1
2
```

## Note
In the first test case, there is only one operation applied to $$$x = 1$$$, resulting in $$$x$$$ becoming $$$2$$$.

In the second test case, for $$$x = 2$$$, within one operation, one is added to $$$x$$$ and it's divided by $$$y = 3$$$, resulting in $$$x$$$ becoming $$$1$$$.

In the third test case, $$$x$$$ changes as follows:

- After the first operation, $$$x = 1$$$, because $$$24 + 1 = 25$$$ and $$$25$$$ is divisible by $$$y = 5$$$ twice within one operation.
- After the second operation, $$$x = 2$$$.
- After the third operation, $$$x = 3$$$.
- After the fourth operation, $$$x = 4$$$.
- After the fifth operation, $$$x = 1$$$.

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
Oh no, the ForceCodes servers are running out of memory! Luckily, you can help them out by uploading some of your RAM!

You want to upload $$$n$$$ GBs of RAM. Every second, you will upload either $$$0$$$ or $$$1$$$ GB of RAM. However, there is a restriction on your network speed: in any $$$k$$$ consecutive seconds, you can upload only at most $$$1$$$ GB of RAM in total.

Find the minimum number of seconds needed to upload $$$n$$$ GBs of RAM!

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

The first and only line of each test case contains two integers $$$n$$$ and $$$k$$$ ($$$1 \le n, k \le 100$$$) — the number of GBs that you want to upload and the length of the time window respectively.

## Output Format
For each test case, output a single integer — the minimum number of seconds needed to upload $$$n$$$ GBs of RAM.

## Examples
```input
6
5 1
2 2
2 3
1 7
11 5
100 100
```
```output
5
3
4
1
51
9901
```

## Note
In the first test case, you can upload $$$1$$$ GB of RAM per second, so to upload $$$5$$$ GBs, you need $$$5$$$ seconds.

In the second test case, you can upload $$$1$$$ GB in the first second, $$$0$$$ GBs in the second second, and $$$1$$$ GB in the third second, which in total adds up to exactly $$$2$$$ GBs of uploaded RAM.

In the third test case, you can upload $$$1$$$ GB in the first second, $$$0$$$ GBs in the second second, $$$0$$$ GBs in the third second, and $$$1$$$ GB in the fourth second, which in total adds up to exactly $$$2$$$ GBs of uploaded RAM.

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 are given an array of integers $$$a$$$ of length $$$n$$$.

You can apply the following operation any number of times (maybe, zero):

- First, choose an integer $$$k$$$ such that $$$1 \le k \le n$$$ and pay $$$k + 1$$$ coins.
- Then, choose exactly $$$k$$$ indices such that $$$1 \le i_1 < i_2 < \ldots < i_k \le n$$$.
- Then, for each $$$x$$$ from $$$1$$$ to $$$k$$$, increase $$$a_{i_x}$$$ by $$$1$$$.

Find the minimum number of coins needed to make $$$a$$$ non-decreasing. That is, $$$a_1 \le a_2 \le \ldots \le a_n$$$.

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

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

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

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

## Output Format
For each test case, output a single integer — the minimum number of coins needed to make $$$a$$$ non-decreasing.

## Examples
```input
5
3
1 7 9
5
2 1 4 7 6
4
1 3 2 4
1
179
9
344 12 37 60 311 613 365 328 675
```
```output
0
3
2
0
1821
```

## Note
In the first test case, $$$a$$$ is already sorted, so you don't have to spend any coins.

In the second test case, the optimal sequence of operations is:

- Choose $$$k = 2$$$ and the indices $$$2$$$ and $$$5$$$: $$$[ 2, \color{red}{1}, 4, 7, \color{red}{6} ] \rightarrow [2, 2, 4, 7, 7]$$$. This costs $$$3$$$ coins.

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
There are $$$n$$$ flowers in a row, the $$$i$$$-th of them initially has a positive height of $$$h_i$$$ meters.

Every second, the wind will blow from the left, causing the height of some flowers to decrease.

Specifically, every second, for each $$$i$$$ from $$$1$$$ to $$$n$$$, in this order, the following happens:

- If $$$i = n$$$ or $$$h_i > h_{i + 1}$$$, the value of $$$h_i$$$ changes to $$$\max(0, h_i - 1)$$$.

How many seconds will pass before $$$h_i=0$$$ for all $$$1 \le i \le n$$$ for the first time?

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

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

The second line of each test case contains $$$n$$$ integers $$$h_1, h_2, \ldots, h_n$$$ ($$$1 \le h_i \le 10 ^ 9$$$) — the heights of the flowers.

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

## Output Format
For each test case, output a single integer — the number of seconds that will pass before $$$h_i=0$$$ for all $$$1 \le i \le n$$$.

## Examples
```input
4
3
1 1 2
2
3 1
1
9
5
7 4 4 3 2
```
```output
4
3
9
7
```

## Note
In the first test case, the flower heights change as follows: $$$[1, 1, 2] \rightarrow [1, 1, 1] \rightarrow [1, 1, 0] \rightarrow [1, 0, 0] \rightarrow [0, 0, 0]$$$.

In the second test case, the flower heights change as follows: $$$[3, 1] \rightarrow [2, 0] \rightarrow [1, 0] \rightarrow [0, 0]$$$.

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
Alice and Bob are playing a game. Initially, there are $$$n$$$ cakes, with the $$$i$$$-th cake having a tastiness value of $$$a_i$$$.

Alice and Bob take turns eating them, with Alice starting first:

- In her turn, Alice chooses and eats any remaining cake whose tastiness is strictly greater than the maximum tastiness of any of the cakes she's eaten before that. Note that on the first turn, she can choose any cake.
- In his turn, Bob chooses any remaining cake and eats it.

The game ends when the current player can't eat a suitable cake. Let $$$x$$$ be the number of cakes that Alice ate. Then, Alice wants to maximize $$$x$$$, while Bob wants to minimize $$$x$$$.

Find out how many cakes Alice will eat if both players play optimally.

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

The first line of each test case contains a single integer $$$n$$$ ($$$1 \le n \le 5000$$$) — the number of cakes.

The second line of each test case contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \le a_i \le n$$$) — the tastiness values of the cakes.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$5000$$$.

## Output Format
For each test case, output a single integer — the number of cakes Alice will eat if both players play optimally.

## Examples
```input
9
4
1 4 2 3
3
1 1 1
5
1 4 2 3 4
4
3 4 1 4
1
1
8
4 3 2 5 6 8 3 4
7
6 1 1 3 5 3 1
11
6 11 6 8 7 5 3 11 2 3 5
17
2 6 5 3 9 1 6 2 5 6 3 2 3 9 6 1 6
```
```output
2
1
3
2
1
3
2
4
4
```

## Note
In the first test case, one possible sequence of turns is:

1. Alice eats a cake with a tastiness value of $$$1$$$. The remaining cakes are $$$[4, 2, 3]$$$.
2. Bob eats a cake with a tastiness value of $$$2$$$. The remaining cakes are $$$[4, 3]$$$.
3. Alice eats a cake with a tastiness of $$$3$$$. The remaining cakes are $$$[4]$$$.
4. Bob eats a cake with a tastiness value of $$$4$$$. The remaining cakes are $$$[]$$$.
5. Since there are no more cakes left, the game ends.

In the second test case, one possible sequence of turns is:

1. Alice eats a cake with a tastiness value of $$$1$$$. The remaining cakes are $$$[1, 1]$$$.
2. Bob eats a cake with a tastiness value of $$$1$$$. The remaining cakes are $$$[1]$$$.
3. Since Alice has already eaten a cake with a tastiness value of $$$1$$$, she cannot make a turn, so the game ends.

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
This is the easy version of the problem. The only difference between the two versions is the constraint on $$$n$$$. You can make hacks only if both versions of the problem are solved.

You are given an array of integers $$$a$$$ of length $$$n$$$.

In one operation, you will perform the following two-step process:

1. Choose an index $$$i$$$ such that $$$1 \le i < |a|$$$ and $$$a_i = i$$$.
2. Remove $$$a_i$$$ and $$$a_{i+1}$$$ from the array and concatenate the remaining parts.

Find the maximum number of times that you can perform the operation above.

## Input Format
Each test contains multiple test cases. The first line of input contains a single integer $$$t$$$ ($$$1 \le t \le 100$$$) — the number of test cases. The description of the test cases follows.

The first line of each test case contains a single integer $$$n$$$ ($$$1 \le n \le 100$$$) — the length of the array $$$a$$$.

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

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$100$$$.

## Output Format
For each test case, output a single integer — the maximum number of times that you can perform the operation.

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

## Note
In the first test case, one possible optimal sequence of operations is $$$[ 1, 5, \color{red}{3}, \color{red}{2}, 4 ] \rightarrow [\color{red}{1}, \color{red}{5}, 4] \rightarrow [4]$$$.

In the third test case, one possible optimal sequence of operations is $$$[1, \color{red}{2}, \color{red}{3}] \rightarrow [1]$$$.

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: 2.0 seconds
Memory limit: 256.0 MB

# Problem
A renowned abstract artist Sasha, drawing inspiration from nowhere, decided to paint a picture entitled "Special Olympics". He justly thought that, if the regular Olympic games have five rings, then the Special ones will do with exactly two rings just fine.

Let us remind you that a ring is a region located between two concentric circles with radii r and R (r < R). These radii are called internal and external, respectively. Concentric circles are circles with centers located at the same point.

Soon a white canvas, which can be considered as an infinite Cartesian plane, had two perfect rings, painted with solid black paint. As Sasha is very impulsive, the rings could have different radii and sizes, they intersect and overlap with each other in any way. We know only one thing for sure: the centers of the pair of rings are not the same.

When Sasha got tired and fell into a deep sleep, a girl called Ilona came into the room and wanted to cut a circle for the sake of good memories. To make the circle beautiful, she decided to cut along the contour.

We'll consider a contour to be a continuous closed line through which there is transition from one color to another (see notes for clarification). If the contour takes the form of a circle, then the result will be cutting out a circle, which Iona wants.

But the girl's inquisitive mathematical mind does not rest: how many ways are there to cut a circle out of the canvas?

## Input Format
The input contains two lines.

Each line has four space-separated integers xi, yi, ri, Ri, that describe the i-th ring; xi and yi are coordinates of the ring's center, ri and Ri are the internal and external radii of the ring correspondingly ( - 100 ≤ xi, yi ≤ 100; 1 ≤ ri < Ri ≤ 100).

It is guaranteed that the centers of the rings do not coinside.

## Output Format
A single integer — the number of ways to cut out a circle from the canvas.

## Examples
```input
60 60 45 55
80 80 8 32
```
```output
1
```
-----
```input
60 60 45 55
80 60 15 25
```
```output
4
```
-----
```input
50 50 35 45
90 50 35 45
```
```output
0
```

## Note
Figures for test samples are given below. The possible cuts are marked with red dotted line.

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 Little Elephant has found a ragged old black-and-white string s on the attic.

The characters of string s are numbered from the left to the right from 1 to |s|, where |s| is the length of the string. Let's denote the i-th character of string s as si. As the string is black-and-white, each character of the string is either letter "B", or letter "W". Unfortunately, the string is very old and some characters are damaged. The damaged positions are denoted as "X".

The Little Elephant in determined to restore the string and hang it on the wall. For that he needs to replace each character "X" by a "B" or a "W". The string must look good on the wall, so it must be beautiful. The Little Elephant considers a string beautiful if it has two non-intersecting substrings of a given length k, such that the left one fully consists of characters "B", and the right one fully consists of characters "W". More formally, there are four integers a, b, c, d (1 ≤ a ≤ b < c ≤ d ≤ |s|; b - a + 1 = d - c + 1 = k) such that si = "B" (a ≤ i ≤ b) and sj = "W" (c ≤ j ≤ d).

Help the Little Elephant find the number of different beautiful strings he can obtain from string s. Two strings are considered different if there is such position, where the character in the first string differs from the corresponding character in the second string. If this string doesn't contain characters «X» and it is already beautiful — the answer is 1.

As the answer can be rather large, print it modulo 1000000007 (109 + 7).

## Input Format
The first line contains two space-separated integers n and k (1 ≤ k ≤ n ≤ 106). The second line contains string s. String s has length n and only consists of characters "W", "B" and "X".

## Output Format
On a single line print an integer — the answer to the problem modulo 1000000007 (109 + 7).

## Examples
```input
3 2
XXX
```
```output
0
```
-----
```input
4 2
XXXX
```
```output
1
```
-----
```input
10 2
XXBXXWXXXX
```
```output
166
```

Now solve the problem and return the code.