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: 1.0 seconds
Memory limit: 256.0 MB

# Problem
Given an array $$$a=[a_1,a_2,\dots,a_n]$$$ of $$$n$$$ positive integers, you can do operations of two types on it:

1. Add $$$1$$$ to every element with an odd index. In other words change the array as follows: $$$a_1 := a_1 +1, a_3 := a_3 + 1, a_5 := a_5+1, \dots$$$.
2. Add $$$1$$$ to every element with an even index. In other words change the array as follows: $$$a_2 := a_2 +1, a_4 := a_4 + 1, a_6 := a_6+1, \dots$$$.

Determine if after any number of operations it is possible to make the final array contain only even numbers or only odd numbers. In other words, determine if you can make all elements of the array have the same parity after any number of operations.

Note that you can do operations of both types any number of times (even none). Operations of different types can be performed a different number of times.

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

The first line of each test case contains an integer $$$n$$$ ($$$2 \leq n \leq 50$$$) — the length of the array.

The second line of each test case contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \leq a_i \leq 10^3$$$) — the elements of the array.

Note that after the performed operations the elements in the array can become greater than $$$10^3$$$.

## Output Format
Output $$$t$$$ lines, each of which contains the answer to the corresponding test case. As an answer, output "YES" if after any number of operations it is possible to make the final array contain only even numbers or only odd numbers, and "NO" otherwise.

You can output the answer in any case (for example, the strings "yEs", "yes", "Yes" and "YES" will be recognized as a positive answer).

## Examples
```input
4
3
1 2 1
4
2 2 2 3
4
2 2 2 2
5
1000 1 1000 1 1000
```
```output
YES
NO
YES
YES
```

## Note
For the first test case, we can increment the elements with an even index, obtaining the array $$$[1, 3, 1]$$$, which contains only odd numbers, so the answer is "YES".

For the second test case, we can show that after performing any number of operations we won't be able to make all elements have the same parity, so the answer is "NO".

For the third test case, all elements already have the same parity so the answer is "YES".

For the fourth test case, we can perform one operation and increase all elements at odd positions by $$$1$$$, thus obtaining the array $$$[1001, 1, 1001, 1, 1001]$$$, and all elements become odd so the answer is "YES".

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
Given $$$n$$$ strings, each of length $$$2$$$, consisting of lowercase Latin alphabet letters from 'a' to 'k', output the number of pairs of indices $$$(i, j)$$$ such that $$$i < j$$$ and the $$$i$$$-th string and the $$$j$$$-th string differ in exactly one position.

In other words, count the number of pairs $$$(i, j)$$$ ($$$i < j$$$) such that the $$$i$$$-th string and the $$$j$$$-th string have exactly one position $$$p$$$ ($$$1 \leq p \leq 2$$$) such that $$${s_{i}}_{p} \neq {s_{j}}_{p}$$$.

The answer may not fit into 32-bit integer type, so you should use 64-bit integers like long long in C++ to avoid integer overflow.

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

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

Then follows $$$n$$$ lines, the $$$i$$$-th of which containing a single string $$$s_i$$$ of length $$$2$$$, consisting of lowercase Latin letters from 'a' to 'k'.

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

## Output Format
For each test case, print a single integer — the number of pairs $$$(i, j)$$$ ($$$i < j$$$) such that the $$$i$$$-th string and the $$$j$$$-th string have exactly one position $$$p$$$ ($$$1 \leq p \leq 2$$$) such that $$${s_{i}}_{p} \neq {s_{j}}_{p}$$$.

Please note, that the answer for some test cases won't fit into 32-bit integer type, so you should use at least 64-bit integer type in your programming language (like long long for C++).

## Examples
```input
4
6
ab
cb
db
aa
cc
ef
7
aa
bb
cc
ac
ca
bb
aa
4
kk
kk
ab
ab
5
jf
jf
jk
jk
jk
```
```output
5
6
0
6
```

## Note
For the first test case the pairs that differ in exactly one position are: ("ab", "cb"), ("ab", "db"), ("ab", "aa"), ("cb", "db") and ("cb", "cc").

For the second test case the pairs that differ in exactly one position are: ("aa", "ac"), ("aa", "ca"), ("cc", "ac"), ("cc", "ca"), ("ac", "aa") and ("ca", "aa").

For the third test case, the are no pairs satisfying the conditions.

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: 1.0 seconds
Memory limit: 256.0 MB

# Problem
There is a grid with $$$n$$$ rows and $$$m$$$ columns, and three types of cells:

- An empty cell, denoted with '.'.
- A stone, denoted with '*'.
- An obstacle, denoted with the lowercase Latin letter 'o'.

All stones fall down until they meet the floor (the bottom row), an obstacle, or other stone which is already immovable. (In other words, all the stones just fall down as long as they can fall.)

Simulate the process. What does the resulting grid look like?

## Input Format
The input consists of multiple test cases. The first line contains an integer $$$t$$$ ($$$1 \leq t \leq 100$$$) — the number of test cases. The description of the test cases follows.

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \leq n, m \leq 50$$$) — the number of rows and the number of columns in the grid, respectively.

Then $$$n$$$ lines follow, each containing $$$m$$$ characters. Each of these characters is either '.', '*', or 'o' — an empty cell, a stone, or an obstacle, respectively.

## Output Format
For each test case, output a grid with $$$n$$$ rows and $$$m$$$ columns, showing the result of the process.

You don't need to output a new line after each test, it is in the samples just for clarity.

## Examples
```input
3
6 10
.*.*....*.
.*.......*
...o....o.
.*.*....*.
..........
.o......o*
2 9
...***ooo
.*o.*o.*o
5 5
*****
*....
*****
....*
*****
```
```output
..........
...*....*.
.*.o....o.
.*........
.*......**
.o.*....o*

....**ooo
.*o**o.*o

.....
*...*
*****
*****
*****
```

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
Let $$$\mathsf{AND}$$$ denote the bitwise AND operation, and $$$\mathsf{OR}$$$ denote the bitwise OR operation.

You are given an array $$$a$$$ of length $$$n$$$ and a non-negative integer $$$k$$$. You can perform at most $$$k$$$ operations on the array of the following type:

- Select an index $$$i$$$ ($$$1 \leq i \leq n$$$) and replace $$$a_i$$$ with $$$a_i$$$ $$$\mathsf{OR}$$$ $$$2^j$$$ where $$$j$$$ is any integer between $$$0$$$ and $$$30$$$ inclusive. In other words, in an operation you can choose an index $$$i$$$ ($$$1 \leq i \leq n$$$) and set the $$$j$$$-th bit of $$$a_i$$$ to $$$1$$$ ($$$0 \leq j \leq 30$$$).

Output the maximum possible value of $$$a_1$$$ $$$\mathsf{AND}$$$ $$$a_2$$$ $$$\mathsf{AND}$$$ $$$\dots$$$ $$$\mathsf{AND}$$$ $$$a_n$$$ after performing at most $$$k$$$ operations.

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

The first line of each test case contains the integers $$$n$$$ and $$$k$$$ ($$$1 \le n \le 2 \cdot 10^5$$$, $$$0 \le k \le 10^9$$$).

Then a single line follows, containing $$$n$$$ integers describing the arrays $$$a$$$ ($$$0 \leq a_i < 2^{31}$$$).

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 a single line containing the maximum possible $$$\mathsf{AND}$$$ value of $$$a_1$$$ $$$\mathsf{AND}$$$ $$$a_2$$$ $$$\mathsf{AND}$$$ $$$\dots$$$ $$$\mathsf{AND}$$$ $$$a_n$$$ after performing at most $$$k$$$ operations.

## Examples
```input
4
3 2
2 1 1
7 0
4 6 6 28 6 6 12
1 30
0
4 4
3 1 3 1
```
```output
2
4
2147483646
1073741825
```

## Note
For the first test case, we can set the bit $$$1$$$ ($$$2^1$$$) of the last $$$2$$$ elements using the $$$2$$$ operations, thus obtaining the array [$$$2$$$, $$$3$$$, $$$3$$$], which has $$$\mathsf{AND}$$$ value equal to $$$2$$$.

For the second test case, we can't perform any operations so the answer is just the $$$\mathsf{AND}$$$ of the whole array which 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 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: 512.0 MB

# Problem
You are given a string $$$s$$$. You have to determine whether it is possible to build the string $$$s$$$ out of strings aa, aaa, bb and/or bbb by concatenating them. You can use the strings aa, aaa, bb and/or bbb any number of times and in any order.

For example:

- aaaabbb can be built as aa $$$+$$$ aa $$$+$$$ bbb;
- bbaaaaabbb can be built as bb $$$+$$$ aaa $$$+$$$ aa $$$+$$$ bbb;
- aaaaaa can be built as aa $$$+$$$ aa $$$+$$$ aa;
- abab cannot be built from aa, aaa, bb and/or bbb.

## Input Format
The first line contains one integer $$$t$$$ ($$$1 \le t \le 1000$$$) — the number of test cases.

Each test case consists of one line containing the string $$$s$$$ ($$$1 \le |s| \le 50$$$), consisting of characters a and/or b.

## Output Format
For each test case, print YES if it is possible to build the string $$$s$$$. Otherwise, print NO.

You may print each letter in any case (for example, YES, yes, Yes will all be recognized as positive answer, NO, no and nO will all be recognized as negative answer).

## Examples
```input
8
aaaabbb
bbaaaaabbb
aaaaaa
abab
a
b
aaaab
bbaaa
```
```output
YES
YES
YES
NO
NO
NO
NO
YES
```

## Note
The first four test cases of the example are described in the statement.

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
You are given $$$n$$$ points with integer coordinates on a coordinate axis $$$OX$$$. The coordinate of the $$$i$$$-th point is $$$x_i$$$. All points' coordinates are distinct and given in strictly increasing order.

For each point $$$i$$$, you can do the following operation no more than once: take this point and move it by $$$1$$$ to the left or to the right (i..e., you can change its coordinate $$$x_i$$$ to $$$x_i - 1$$$ or to $$$x_i + 1$$$). In other words, for each point, you choose (separately) its new coordinate. For the $$$i$$$-th point, it can be either $$$x_i - 1$$$, $$$x_i$$$ or $$$x_i + 1$$$.

Your task is to determine if you can move some points as described above in such a way that the new set of points forms a consecutive segment of integers, i. e. for some integer $$$l$$$ the coordinates of points should be equal to $$$l, l + 1, \ldots, l + n - 1$$$.

Note that the resulting points should have distinct coordinates.

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. Then $$$t$$$ test cases follow.

The first line of the test case contains one integer $$$n$$$ ($$$1 \le n \le 2 \cdot 10^5$$$) — the number of points in the set $$$x$$$.

The second line of the test case contains $$$n$$$ integers $$$x_1 < x_2 < \ldots < x_n$$$ ($$$1 \le x_i \le 10^6$$$), where $$$x_i$$$ is the coordinate of the $$$i$$$-th point.

It is guaranteed that the points are given in strictly increasing order (this also means that all coordinates are distinct). It is also guaranteed that the sum of $$$n$$$ does not exceed $$$2 \cdot 10^5$$$ ($$$\sum n \le 2 \cdot 10^5$$$).

## Output Format
For each test case, print the answer — if the set of points from the test case can be moved to form a consecutive segment of integers, print YES, otherwise print NO.

## Examples
```input
5
2
1 4
3
1 2 3
4
1 2 3 7
1
1000000
3
2 5 6
```
```output
YES
YES
NO
YES
YES
```

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
You are given a sequence of $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$. You are also given $$$x$$$ integers $$$1, 2, \dots, x$$$.

You are asked to insert each of the extra integers into the sequence $$$a$$$. Each integer can be inserted at the beginning of the sequence, at the end of the sequence, or between any elements of the sequence.

The score of the resulting sequence $$$a'$$$ is the sum of absolute differences of adjacent elements in it $$$\left(\sum \limits_{i=1}^{n+x-1} |a'_i - a'_{i+1}|\right)$$$.

What is the smallest possible score of the resulting sequence $$$a'$$$?

## 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 $$$x$$$ ($$$1 \le n, x \le 2 \cdot 10^5$$$) — the length of the sequence and the number of extra integers.

The second line of each testcase contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 2 \cdot 10^5$$$).

The sum of $$$n$$$ over all testcases doesn't exceed $$$2 \cdot 10^5$$$.

## Output Format
For each testcase, print a single integer — the smallest sum of absolute differences of adjacent elements of the sequence after you insert the extra integers into it.

## Examples
```input
4
1 5
10
3 8
7 2 10
10 2
6 1 5 7 3 3 9 10 10 1
4 10
1 3 1 2
```
```output
9
15
31
13
```

## Note
Here are the sequences with the smallest scores for the example. The underlined elements are the extra integers. Note that there exist other sequences with this smallest score.

- $$$\underline{1}, \underline{2}, \underline{3}, \underline{4}, \underline{5}, 10$$$
- $$$\underline{7}, 7, \underline{6}, \underline{4}, 2, \underline{2}, \underline{1}, \underline{3}, \underline{5}, \underline{8}, 10$$$
- $$$6, \underline{1}, 1, \underline{2}, 5, 7, 3, 3, 9, 10, 10, 1$$$
- $$$1, 3, \underline{1}, 1, 2, \underline{2}, \underline{3}, \underline{4}, \underline{5}, \underline{6}, \underline{7}, \underline{8}, \underline{9}, \underline{10}$$$

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: 512.0 MB

# Problem
You are given a rooted tree of $$$2^n - 1$$$ vertices. Every vertex of this tree has either $$$0$$$ children, or $$$2$$$ children. All leaves of this tree have the same distance from the root, and for every non-leaf vertex, one of its children is the left one, and the other child is the right one. Formally, you are given a perfect binary tree.

The vertices of the tree are numbered in the following order:

- the root has index $$$1$$$;
- if a vertex has index $$$x$$$, then its left child has index $$$2x$$$, and its right child has index $$$2x+1$$$.

Every vertex of the tree has a letter written on it, either A or B. Let's define the character on the vertex $$$x$$$ as $$$s_x$$$.

Let the preorder string of some vertex $$$x$$$ be defined in the following way:

- if the vertex $$$x$$$ is a leaf, then the preorder string of $$$x$$$ be consisting of only one character $$$s_x$$$;
- otherwise, the preorder string of $$$x$$$ is $$$s_x + f(l_x) + f(r_x)$$$, where $$$+$$$ operator defines concatenation of strings, $$$f(l_x)$$$ is the preorder string of the left child of $$$x$$$, and $$$f(r_x)$$$ is the preorder string of the right child of $$$x$$$.

The preorder string of the tree is the preorder string of its root.

Now, for the problem itself...

You have to calculate the number of different strings that can be obtained as the preorder string of the given tree, if you are allowed to perform the following operation any number of times before constructing the preorder string of the tree:

- choose any non-leaf vertex $$$x$$$, and swap its children (so, the left child becomes the right one, and vice versa).

## Input Format
The first line contains one integer $$$n$$$ ($$$2 \le n \le 18$$$).

The second line contains a sequence of $$$2^n-1$$$ characters $$$s_1, s_2, \dots, s_{2^n-1}$$$. Each character is either A or B. The characters are not separated by spaces or anything else.

## Output Format
Print one integer — the number of different strings that can be obtained as the preorder string of the given tree, if you can apply any number of operations described in the statement. Since it can be very large, print it modulo $$$998244353$$$.

## Examples
```input
4
BAAAAAAAABBABAB
```
```output
16
```
-----
```input
2
BAA
```
```output
1
```
-----
```input
2
ABA
```
```output
2
```
-----
```input
2
AAB
```
```output
2
```
-----
```input
2
AAA
```
```output
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 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: 1.0 seconds
Memory limit: 256.0 MB

# Problem
In the pet store on sale there are:

- $$$a$$$ packs of dog food;
- $$$b$$$ packs of cat food;
- $$$c$$$ packs of universal food (such food is suitable for both dogs and cats).

Polycarp has $$$x$$$ dogs and $$$y$$$ cats. Is it possible that he will be able to buy food for all his animals in the store? Each of his dogs and each of his cats should receive one pack of suitable food for it.

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

Then $$$t$$$ lines are given, each containing a description of one test case. Each description consists of five integers $$$a, b, c, x$$$ and $$$y$$$ ($$$0 \le a,b,c,x,y \le 10^8$$$).

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

- YES, if suitable food can be bought for each of $$$x$$$ dogs and for each of $$$y$$$ cats;
- NO else.

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

## Examples
```input
7
1 1 4 2 3
0 0 0 0 0
5 5 0 4 6
1 1 1 1 1
50000000 50000000 100000000 100000000 100000000
0 0 0 100000000 100000000
1 3 2 2 5
```
```output
YES
YES
NO
YES
YES
NO
NO
```

Now solve the problem and return the code.