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: 1.5 seconds
Memory limit: 256.0 MB

# Problem
Petya and his friend, the robot Petya++, have a common friend — the cyborg Petr#. Sometimes Petr# comes to the friends for a cup of tea and tells them interesting problems.

Today, Petr# told them the following problem.

A palindrome is a sequence that reads the same from left to right as from right to left. For example, $$$[38, 12, 8, 12, 38]$$$, $$$[1]$$$, and $$$[3, 8, 8, 3]$$$ are palindromes.

Let's call the palindromicity of a sequence $$$a_1, a_2, \dots, a_n$$$ the minimum count of elements that need to be replaced to make this sequence a palindrome. For example, the palindromicity of the sequence $$$[38, 12, 8, 38, 38]$$$ is $$$1$$$ since it is sufficient to replace the number $$$38$$$ at the fourth position with the number $$$12$$$. And the palindromicity of the sequence $$$[3, 3, 5, 5, 5]$$$ is two since you can replace the first two threes with fives, and the resulting sequence $$$[5, 5, 5, 5, 5]$$$ is a palindrome.

Given a sequence $$$a$$$ of length $$$n$$$, and an odd integer $$$k$$$, you need to find the sum of palindromicity of all subarrays of length $$$k$$$, i. e., the sum of the palindromicity values for the sequences $$$a_i, a_{i+1}, \dots, a_{i+k-1}$$$ for all $$$i$$$ from $$$1$$$ to $$$n-k+1$$$.

The students quickly solved the problem. Can you do it too?

## Input Format
The first line of the input contains two integers $$$n$$$ and $$$k$$$ ($$$1 \le n \le 2 \cdot 10^5$$$, $$$1 \le k \le n$$$, $$$k$$$ is odd) — the length of the sequence and the length of subarrays for which it is necessary to determine whether they are palindromes.

The second line of the input contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \le a_i \le 2 \cdot 10^5$$$) — the sequence itself.

## Output Format
Output a single integer — the total palindromicity of all subarrays of length $$$k$$$.

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

## Note
In the first example, the palindromicity of the subarray $$$[1, 2, 8, 2, 5]$$$ is $$$1$$$, the palindromicity of the string $$$[2, 8, 2, 5, 2]$$$ is also $$$1$$$, the palindromicity of the string $$$[8, 2, 5, 2, 8]$$$ is $$$0$$$, and the palindromicity of the string $$$[2, 5, 2, 8, 6]$$$ is $$$2$$$. The total palindromicity is $$$1+1+0+2 = 4$$$.

In the second example, the only substring of length $$$9$$$ coincides with the entire string, and its palindromicity is $$$0$$$, so the answer is also $$$0$$$.

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: 3.0 seconds
Memory limit: 256.0 MB

# Problem
This is the easy version of the problem. The only difference between the three versions is the constraints on $$$n$$$ and $$$k$$$. You can make hacks only if all versions of the problem are solved.

Maxim is a minibus driver on Venus.

To ride on Maxim's minibus, you need a ticket. Each ticket has a number consisting of $$$n$$$ digits. However, as we know, the residents of Venus use a numeral system with base $$$k$$$, rather than the decimal system. Therefore, the ticket number can be considered as a sequence of $$$n$$$ integers from $$$0$$$ to $$$k-1$$$, inclusive.

The residents of Venus consider a ticket to be lucky if there is a digit on it that is equal to the sum of the remaining digits, modulo $$$k$$$. For example, if $$$k=10$$$, then the ticket $$$7135$$$ is lucky because $$$7 + 1 + 5 \equiv 3 \pmod{10}$$$. On the other hand, the ticket $$$7136$$$ is not lucky because no digit is equal to the sum of the others modulo $$$10$$$.

Once, while on a trip, Maxim wondered: how many lucky tickets exist? At the same time, Maxim understands that this number can be very large, so he is interested only in the answer modulo some prime number $$$m$$$.

## Input Format
The only line of the input contains three integers $$$n$$$, $$$k$$$ and $$$m$$$ ($$$1 \le n \le 100$$$, $$$1 \le k \le 30$$$, $$$10^8 \le m \le 10^9 + 7$$$, $$$m$$$ is a prime number) — the number of digits on the ticket, the base of the numeral system on Venus, and the module for answer calculation.

## Output Format
Print one integer — the number of lucky tickets modulo $$$m$$$, i. e. the remainder after dividing the answer by $$$m$$$.

## Examples
```input
3 2 1000000007
```
```output
4
```
-----
```input
3 4 1000000007
```
```output
28
```

## Note
In the first example, there are only four lucky tickets: $$$000$$$, $$$011$$$, $$$101$$$, and $$$110$$$.

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
This is the medium version of the problem. The only difference between the three versions is the constraints on $$$n$$$ and $$$k$$$. You can make hacks only if all versions of the problem are solved.

Maxim is a minibus driver on Venus.

To ride on Maxim's minibus, you need a ticket. Each ticket has a number consisting of $$$n$$$ digits. However, as we know, the residents of Venus use a numeral system with base $$$k$$$, rather than the decimal system. Therefore, the ticket number can be considered as a sequence of $$$n$$$ integers from $$$0$$$ to $$$k-1$$$, inclusive.

The residents of Venus consider a ticket to be lucky if there is a digit on it that is equal to the sum of the remaining digits, modulo $$$k$$$. For example, if $$$k=10$$$, then the ticket $$$7135$$$ is lucky because $$$7 + 1 + 5 \equiv 3 \pmod{10}$$$. On the other hand, the ticket $$$7136$$$ is not lucky because no digit is equal to the sum of the others modulo $$$10$$$.

Once, while on a trip, Maxim wondered: how many lucky tickets exist? At the same time, Maxim understands that this number can be very large, so he is interested only in the answer modulo some prime number $$$m$$$.

## Input Format
The only line of the input contains three integers $$$n$$$, $$$k$$$ and $$$m$$$ ($$$1 \le n \le 10^{18}$$$, $$$1 \le k \le 100$$$, $$$10^8 \le m \le 10^9 + 7$$$, $$$m$$$ is a prime number) — the number of digits on the ticket, the base of the numeral system on Venus, and the module for answer calculation.

## Output Format
Print one integer — the number of lucky tickets modulo $$$m$$$, i. e. the remainder after dividing the answer by $$$m$$$.

## Examples
```input
3 2 1000000007
```
```output
4
```
-----
```input
3 4 1000000007
```
```output
28
```

## Note
In the first example, there are only four lucky tickets: $$$000$$$, $$$011$$$, $$$101$$$, and $$$110$$$.

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
This is the hard version of the problem. The only difference between the three versions is the constraints on $$$n$$$ and $$$k$$$. You can make hacks only if all versions of the problem are solved.

Maxim is a minibus driver on Venus.

To ride on Maxim's minibus, you need a ticket. Each ticket has a number consisting of $$$n$$$ digits. However, as we know, the residents of Venus use a numeral system with base $$$k$$$, rather than the decimal system. Therefore, the ticket number can be considered as a sequence of $$$n$$$ integers from $$$0$$$ to $$$k-1$$$, inclusive.

The residents of Venus consider a ticket to be lucky if there is a digit on it that is equal to the sum of the remaining digits, modulo $$$k$$$. For example, if $$$k=10$$$, then the ticket $$$7135$$$ is lucky because $$$7 + 1 + 5 \equiv 3 \pmod{10}$$$. On the other hand, the ticket $$$7136$$$ is not lucky because no digit is equal to the sum of the others modulo $$$10$$$.

Once, while on a trip, Maxim wondered: how many lucky tickets exist? At the same time, Maxim understands that this number can be very large, so he is interested only in the answer modulo some prime number $$$m$$$.

## Input Format
The only line of the input contains three integers $$$n$$$, $$$k$$$ and $$$m$$$ ($$$1 \le n \le 10^{18}$$$, $$$1 \le k \le 2000$$$, $$$10^8 \le m \le 10^9 + 7$$$, $$$m$$$ is a prime number) — the number of digits on the ticket, the base of the numeral system on Venus, and the module for answer calculation.

## Output Format
Print one integer — the number of lucky tickets modulo $$$m$$$, i. e. the remainder after dividing the answer by $$$m$$$.

## Examples
```input
3 2 1000000007
```
```output
4
```
-----
```input
3 4 1000000007
```
```output
28
```

## Note
In the first example, there are only four lucky tickets: $$$000$$$, $$$011$$$, $$$101$$$, and $$$110$$$.

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
In Berland, there are two types of coins, having denominations of $$$2$$$ and $$$k$$$ burles.

Your task is to determine whether it is possible to represent $$$n$$$ burles in coins, i. e. whether there exist non-negative integers $$$x$$$ and $$$y$$$ such that $$$2 \cdot x + k \cdot y = n$$$.

## 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 each test case contains two integers $$$n$$$ and $$$k$$$ ($$$1 \le k \le n \le 10^{18}$$$; $$$k \ne 2$$$).

## Output Format
For each test case, print YES if it is possible to represent $$$n$$$ burles in coins; otherwise, print NO. You may print each letter in any case (YES, yes, Yes will all be recognized as positive answer, NO, no and nO will all be recognized as negative answer).

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

## Note
In the first test case, you can take one coin with denomination $$$2$$$ and one coin with denomination $$$k = 3$$$.

In the second test case, you can take three coins with denomination $$$2$$$. Alternatively, you can take six coins with denomination $$$k = 1$$$.

In the third test case, there is no way to represent $$$7$$$ burles.

In the fourth test case, you can take one coin with denomination $$$k = 8$$$.

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: 2.0 seconds
Memory limit: 256.0 MB

# Problem
A robot is placed in a cell $$$(0, 0)$$$ of an infinite grid. This robot has adjustable length legs. Initially, its legs have length $$$1$$$.

Let the robot currently be in the cell $$$(x, y)$$$ and have legs of length $$$m$$$. In one move, it can perform one of the following three actions:

- jump into the cell $$$(x + m, y)$$$;
- jump into the cell $$$(x, y + m)$$$;
- increase the length of the legs by $$$1$$$, i. e. set it to $$$m + 1$$$.

What's the smallest number of moves robot has to make to reach a cell $$$(a, b)$$$?

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

The only line of each test case contains two integers $$$a$$$ and $$$b$$$ ($$$1 \le a, b \le 10^9$$$) — the ending cell.

## Output Format
For each test case, print a single integer — the smallest number of moves the robot is required to make to reach a cell $$$(a, b)$$$ from a cell $$$(0, 0)$$$.

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

## Note
In the first testcase, the robot can first jump to $$$(0, 1)$$$, then to $$$(1, 1)$$$. If it ever increases the length of its legs, it will only be able to jump past $$$(1, 1)$$$.

In the second testcase, the robot can jump to $$$(1, 0)$$$, then increase the length of its length to $$$2$$$ and jump three times to reach $$$(1, 6)$$$.

In the third testcase, the robot can increase the length of its legs three times to make it $$$4$$$. Then jump to $$$(0, 4)$$$. Then jump twice to reach $$$(8, 4)$$$.

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: 3.0 seconds
Memory limit: 512.0 MB

# Problem
You are given an undirected graph consisting of $$$n$$$ vertices and $$$n-1$$$ edges. The $$$i$$$-th edge has weight $$$a_i$$$; it connects the vertices $$$i$$$ and $$$i+1$$$.

Initially, each vertex contains a chip. Each chip has an integer written on it; the integer written on the chip in the $$$i$$$-th vertex is $$$i$$$.

In one operation, you can choose a chip (if there are multiple chips in a single vertex, you may choose any one of them) and move it along one of the edges of the graph. The cost of this operation is equal to the weight of the edge.

The cost of the graph is the minimum cost of a sequence of such operations that meets the following condition:

- after all operations are performed, each vertex contains exactly one chip, and the integer on each chip is not equal to the index of the vertex where that chip is located.

You are given $$$q$$$ queries of the form:

- $$$k$$$ $$$x$$$ — change the weight of the $$$k$$$-th edge (the one which connects the vertices $$$k$$$ and $$$k+1$$$) to $$$x$$$.

After each query, print the cost of the graph. Note that you don't actually move any chips; when you compute the cost, the chips are on their initial positions.

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

The second line contains $$$n-1$$$ integers $$$a_1, a_2, \dots, a_{n-1}$$$ ($$$1 \le a_i \le 10^9$$$).

The third line contains one integer $$$q$$$ ($$$1 \le q \le 2 \cdot 10^5$$$).

Then $$$q$$$ lines follow. The $$$i$$$-th of them contains two integers $$$k$$$ and $$$x$$$ ($$$1 \le k \le n-1$$$; $$$1 \le x \le 10^9$$$) for the $$$i$$$-th query.

## Output Format
For each query, print one integer — the cost of the graph after the query is performed.

## Examples
```input
10
12 6 12 15 20 8 17 12 15
8
4 10
7 3
6 14
9 9
2 10
3 5
4 11
7 11
```
```output
126
108
120
108
112
98
98
114
```

Now solve the problem and return the code.