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: 256.0 MB

# Problem
You are given an array $$$a_1, a_2, \ldots, a_n$$$. Find the number of tuples ($$$x, y, z$$$) such that:

- $$$1 \leq x \leq y \leq z \leq n$$$, and
- $$$f(x, y) \oplus f(y, z) > f(x, z)$$$.

We define $$$f(l, r) = a_l \oplus a_{l + 1} \oplus \ldots \oplus a_{r}$$$, where $$$\oplus$$$ denotes the bitwise XOR operation.

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

The first line of each test case contains a single integer $$$n$$$ ($$$1 \leq n \leq 10^5$$$).

The second line of each test case contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \leq a_i \leq 10^9$$$).

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 on a new line — the number of described tuples.

## Examples
```input
3
3
6 2 4
1
3
5
7 3 7 2 1
```
```output
4
0
16
```

## Note
In the first case, there are 4 such tuples in the array $$$[6, 2, 4]$$$:

- ($$$1$$$, $$$2$$$, $$$2$$$): $$$(a_1 \oplus a_2) \oplus (a_2) = 4 \oplus 2 > (a_1 \oplus a_2) = 4$$$
- ($$$1$$$, $$$1$$$, $$$3$$$): $$$(a_1) \oplus (a_1 \oplus a_2 \oplus a_3) = 6 \oplus 0 > (a_1 \oplus a_2 \oplus a_3) = 0$$$
- ($$$1$$$, $$$2$$$, $$$3$$$): $$$(a_1 \oplus a_2) \oplus (a_2 \oplus a_3) = 4 \oplus 6 > (a_1 \oplus a_2 \oplus a_3) = 0$$$
- ($$$1$$$, $$$3$$$, $$$3$$$): $$$(a_1 \oplus a_2 \oplus a_3) \oplus (a_3) = 0 \oplus 4 > (a_1 \oplus a_2 \oplus a_3) = 0$$$

In the second test case, there are no such tuples.

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: 2.5 seconds
Memory limit: 256.0 MB

# Problem
You are given an integer $$$n$$$. The function $$$C(i,k)$$$ represents the number of distinct ways you can select $$$k$$$ distinct numbers from the set {$$$1, 2, \ldots, i$$$} and arrange them in a circle$$$^\dagger$$$.

Find the value of $$$$$$ \sum\limits_{i=1}^n \sum\limits_{j=1}^i \left( C(i,j) \bmod j \right). $$$$$$ Here, the operation $$$x \bmod y$$$ denotes the remainder from dividing $$$x$$$ by $$$y$$$.

Since this value can be very large, find it modulo $$$10^9+7$$$.

$$$^\dagger$$$ In a circular arrangement, sequences are considered identical if one can be rotated to match the other. For instance, $$$[1, 2, 3]$$$ and $$$[2, 3, 1]$$$ are equivalent in a circle.

## Input Format
The first line contains a single integer $$$t$$$ ($$$1 \leq t \leq 10^5$$$) — the number of test cases.

The only line of each test case contains a single integer $$$n$$$ ($$$1 \le n \le 10^6$$$).

## Output Format
For each test case, output a single integer on a new line — the value of the expression to be calculated modulo $$$10^9+7$$$.

## Examples
```input
4
1
3
6
314159
```
```output
0
4
24
78926217
```

## Note
In the first test case, $$$C(1,1) \bmod 1 = 0$$$.

In the second test case:

- $$$C(1,1)=1$$$ (the arrangements are: $$$[1]$$$);
- $$$C(2,1)=2$$$ (the arrangements are: $$$[1]$$$, $$$[2]$$$);
- $$$C(2,2)=1$$$ (the arrangements are: $$$[1, 2]$$$);
- $$$C(3,1)=3$$$ (the arrangements are: $$$[1]$$$, $$$[2]$$$, $$$[3]$$$);
- $$$C(3,2)=3$$$ (the arrangements are: $$$[1, 2]$$$, $$$[2, 3]$$$, $$$[3, 1]$$$);
- $$$C(3,3)=2$$$ (the arrangements are: $$$[1, 2, 3]$$$, $$$[1, 3, 2]$$$).

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: 4.0 seconds
Memory limit: 512.0 MB

# Problem
This is the easy version of the problem. The difference between the two versions of this problem is the constraint on $$$k$$$. You can make hacks only if all versions of the problem are solved.

You are given an undirected tree of $$$n$$$ nodes. Each node $$$v$$$ has a value $$$a_v$$$ written on it. You have to answer queries related to the tree.

You are given $$$q$$$ queries. In each query, you are given $$$5$$$ integers, $$$u_1, v_1, u_2, v_2, k$$$. Denote the count of nodes with value $$$c$$$ on path $$$u_1 \rightarrow v_1$$$ with $$$x_c$$$, and the count of nodes with value $$$c$$$ on path $$$u_2 \rightarrow v_2$$$ with $$$y_c$$$. If there are $$$z$$$ such values of $$$c$$$ such that $$$x_c \neq y_c$$$, output any $$$\min(z, k)$$$ such values in any order.

## Input Format
The first line contains one integer $$$n$$$ ($$$1 \leq n \leq 10^5$$$) — the number of nodes in the tree.

The next line contains $$$n$$$ integers, $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \leq a_i \leq 10^5$$$) — the value written on each node of the tree.

Then $$$n - 1$$$ lines follow. Each line contains two integers $$$u$$$ and $$$v$$$ ($$$1 \leq u, v \leq n, u \neq v$$$) denoting an edge of the tree. It is guaranteed that the given edges form a tree.

The next line contains one integer $$$q$$$ ($$$1 \leq q \leq 10^5$$$) — the number of queries.

Then $$$q$$$ lines follow. Each line contains five integers $$$u_1, v_1, u_2, v_2, k$$$ ($$$1 \leq u_1, v_1, u_2, v_2 \leq n$$$, $$$k = 1$$$).

## Output Format
For each query, output on a separate line. For a query, first output $$$\min(z, k)$$$ and then on the same line, output any $$$\min(z, k)$$$ values in any order which occur a different number of times in each path.

## Examples
```input
5
5 2 3 4 3
1 2
1 3
2 4
2 5
3
1 4 4 5 1
2 3 2 3 1
5 5 4 3 1
```
```output
1 5
0
1 2
```

## Note
For query $$$1$$$, the first path is $$$1 \rightarrow 2 \rightarrow 4$$$, coming across the multiset of values $$$\{5, 2, 4\}$$$. On the second path $$$4 \rightarrow 2 \rightarrow 5$$$, we have the multiset $$$\{4, 2, 3\}$$$. Two numbers — $$$3$$$ and $$$5$$$ occur a different number of times, hence we print one of them.

In query $$$2$$$, there is no difference between the paths, hence we output $$$0$$$.

In query $$$3$$$, the first path is just the node $$$5$$$, resulting in the multiset $$$\{3\}$$$, and the second path $$$4 \rightarrow 2 \rightarrow 1 \rightarrow 3$$$ gives $$$\{4, 2, 5, 3\}$$$. The numbers $$$5$$$, $$$2$$$ and $$$4$$$ occur a different number of 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: 1.0 seconds
Memory limit: 256.0 MB

# Problem
You have a hand of $$$n$$$ cards, where each card has a number written on it, and a fixed integer $$$k$$$. You can perform the following operation any number of times:

- Choose any $$$k$$$ cards from your hand that all have the same number.
- Exchange these cards for $$$k-1$$$ cards, each of which can have any number you choose (including the number written on the cards you just exchanged).

Here is one possible sequence of operations for the first example case, which has $$$k=3$$$:

What is the minimum number of cards you can have in your hand at the end of this process?

## Input Format
The first line of the 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 two integers $$$n$$$ and $$$k$$$ ($$$1 \le n \le 100$$$, $$$2 \le k \le 100$$$) — the number of cards you have, and the number of cards you exchange during each operation, respectively.

The next line of each test case contains $$$n$$$ integers $$$c_1, c_2, \ldots c_n$$$ ($$$1 \le c_i \le 100$$$) — the numbers written on your cards.

## Output Format
For each test case, output a single integer — the minimum number of cards you can have left in your hand after any number of operations.

## Examples
```input
7
5 3
4 1 1 4 4
1 10
7
7 2
4 2 1 100 5 2 3
10 4
1 1 1 1 1 1 1 1 1 1
5 2
3 8 1 48 7
6 2
10 20 30 10 20 40
6 3
10 20 30 10 20 40
```
```output
2
1
1
3
5
1
6
```

## Note
The first example case corresponds to the picture above. The sequence of operations displayed there is optimal, so the answer is $$$2$$$.

In the second example case, no operations can be performed, so the answer is $$$1$$$.

In the fourth example case, you can repeatedly select $$$4$$$ cards numbered with $$$1$$$ and replace them with $$$3$$$ cards numbered with $$$1$$$, until there are $$$3$$$ cards left.

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
There is an $$$n \times m$$$ grid of white and black squares. In one operation, you can select any two squares of the same color, and color all squares in the subrectangle between them that color.

Formally, if you select positions $$$(x_1, y_1)$$$ and $$$(x_2, y_2)$$$, both of which are currently the same color $$$c$$$, set the color of all $$$(x, y)$$$ where $$$\min(x_1, x_2) \le x \le \max(x_1, x_2)$$$ and $$$\min(y_1, y_2) \le y \le \max(y_1, y_2)$$$ to $$$c$$$.

This diagram shows a sequence of two possible operations on a grid:

Is it possible for all squares in the grid to be the same color, after performing any number of operations (possibly zero)?

## Input Format
The first line of the 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 two integers $$$n$$$ and $$$m$$$ ($$$1 \le n, m \le 500$$$) — the number of rows and columns in the grid, respectively.

Each of the next $$$n$$$ lines contains $$$m$$$ characters 'W' and 'B' — the initial colors of the squares of the grid.

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

## Output Format
For each test case, print "YES" if it is possible to make all squares in the grid the same color, and "NO" otherwise.

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
8
2 1
W
B
6 6
WWWWBW
WBWWWW
BBBWWW
BWWWBB
WWBWBB
BBBWBW
1 1
W
2 2
BB
BB
3 4
BWBW
WBWB
BWBW
4 2
BB
BB
WW
WW
4 4
WWBW
BBWB
WWBB
BBBB
1 5
WBBWB
```
```output
NO
YES
YES
YES
YES
NO
YES
NO
```

## Note
In the first example, it is impossible to ever change the color of any square with an operation, so we output NO.

The second example is the case pictured above. As shown in that diagram, it is possible for all squares to be white after two operations, so we output YES.

In the third and fourth examples, all squares are already the same color, so we output YES.

In the fifth example we can do everything in two operations. First, select positions $$$(2, 1)$$$ and $$$(1, 4)$$$ and color all squares with $$$1 \le x \le 2$$$ and $$$1 \le y \le 4$$$ to white. Then, select positions $$$(2, 1)$$$ and $$$(3, 4)$$$ and color all squares with $$$2 \le x \le 3$$$ and $$$1 \le y \le 4$$$ to white. After these two operations all squares are white.

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
A contest contains $$$n$$$ problems and the difficulty of the $$$i$$$-th problem is expected to be at most $$$b_i$$$. There are already $$$n$$$ problem proposals and the difficulty of the $$$i$$$-th problem is $$$a_i$$$. Initially, both $$$a_1, a_2, \ldots, a_n$$$ and $$$b_1, b_2, \ldots, b_n$$$ are sorted in non-decreasing order.

Some of the problems may be more difficult than expected, so the writers must propose more problems. When a new problem with difficulty $$$w$$$ is proposed, the most difficult problem will be deleted from the contest, and the problems will be sorted in a way that the difficulties are non-decreasing.

In other words, in each operation, you choose an integer $$$w$$$, insert it into the array $$$a$$$, sort array $$$a$$$ in non-decreasing order, and remove the last element from it.

Find the minimum number of new problems to make $$$a_i\le b_i$$$ for all $$$i$$$.

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

The first line of each test case contains only one positive integer $$$n$$$ ($$$1 \leq n \leq 100$$$), representing the number of problems.

The second line of each test case contains an array $$$a$$$ of length $$$n$$$ ($$$1\le a_1\le a_2\le\cdots\le a_n\le 10^9$$$).

The third line of each test case contains an array $$$b$$$ of length $$$n$$$ ($$$1\le b_1\le b_2\le\cdots\le b_n\le 10^9$$$).

## Output Format
For each test case, print an integer as your answer in a new line.

## Examples
```input
2
6
1000 1400 2000 2000 2200 2700
800 1200 1500 1800 2200 3000
6
4 5 6 7 8 9
1 2 3 4 5 6
```
```output
2
3
```

## Note
In the first test case:

- Propose a problem with difficulty $$$w=800$$$ and $$$a$$$ becomes $$$[800,1000,1400,2000,2000,2200]$$$.
- Propose a problem with difficulty $$$w=1800$$$ and $$$a$$$ becomes $$$[800,1000,1400,1800,2000,2000]$$$.

It can be proved that it's impossible to reach the goal by proposing fewer new problems.

In the second test case:

- Propose a problem with difficulty $$$w=1$$$ and $$$a$$$ becomes $$$[1,4,5,6,7,8]$$$.
- Propose a problem with difficulty $$$w=2$$$ and $$$a$$$ becomes $$$[1,2,4,5,6,7]$$$.
- Propose a problem with difficulty $$$w=3$$$ and $$$a$$$ becomes $$$[1,2,3,4,5,6]$$$.

It can be proved that it's impossible to reach the goal by proposing fewer new problems.

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
Given two arrays of distinct positive integers $$$a$$$ and $$$b$$$ of length $$$n$$$, we would like to make both the arrays the same. Two arrays $$$x$$$ and $$$y$$$ of length $$$k$$$ are said to be the same when for all $$$1 \le i \le k$$$, $$$x_i = y_i$$$.

Now in one move, you can choose some index $$$l$$$ and $$$r$$$ in $$$a$$$ ($$$l \le r$$$) and swap $$$a_l$$$ and $$$a_r$$$, then choose some $$$p$$$ and $$$q$$$ ($$$p \le q$$$) in $$$b$$$ such that $$$r-l=q-p$$$ and swap $$$b_p$$$ and $$$b_q$$$.

Is it possible to make both arrays the same?

## Input Format
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 2 \cdot 10^4$$$). 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 arrays $$$a$$$ and $$$b$$$.

The second line of each test case contains $$$n$$$ distinct integers $$$a_1,a_2,a_3,\ldots,a_n$$$ ($$$1 \le a_i \le 2 \cdot 10^5$$$) — the integers in the array $$$a$$$.

The third line of each test case contains $$$n$$$ distinct integers $$$b_1,b_2,b_3,\ldots,b_n$$$ ($$$1 \le b_i \le 2 \cdot 10^5$$$) — the integers in the array $$$b$$$.

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

## Output Format
For each testcase, print "YES" if the arrays $$$a$$$ and $$$b$$$ can be made the same. Otherwise, print "NO". 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
6
4
1 2 3 4
1 2 3 4
5
1 3 4 2 5
7 1 2 5 4
4
1 2 3 4
4 3 2 1
3
1 2 3
1 3 2
5
1 5 7 1000 4
4 1 7 5 1000
3
1 4 2
1 3 2
```
```output
YES
NO
YES
NO
NO
NO
```

## Note
In the first testcase, you don't need to perform any operations since the arrays are same.

In the second testcase, it can be proven there exists no way to make the arrays same.

In the third testcase, one of the ways to make the arrays same is to first choose $$$l=1$$$, $$$r=3$$$, $$$p=1$$$, $$$q=3$$$ then choose $$$l=1$$$, $$$r=2$$$, $$$p=3$$$, $$$q=4$$$.

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: 4.0 seconds
Memory limit: 256.0 MB

# Problem
You have an array of non-negative integers $$$a_1, a_2, \ldots, a_n$$$.

The value of a sub-array of length $$$\ge 2$$$, $$$a[l, r] = [a_l, a_{l+1}, \ldots, a_r]$$$ is the minimum value of $$$a_i \oplus a_j$$$ such that $$$l \le i < j \le r$$$, where $$$\oplus$$$ is the xor (exclusive-or) operator.

You have to find the $$$k$$$-th smallest value over all sub-arrays of length $$$\ge 2$$$.

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

The first line of each test case contains integer numbers $$$n$$$ and $$$k$$$ ($$$2 \le n \le 10^5$$$, $$$1 \le k \le \frac{n\cdot(n-1)}{2}$$$).

The second line of the input contains $$$n$$$ non-negative integer numbers $$$a_1, a_2, \ldots, a_n$$$ ($$$0 \le a_i \le 10^9$$$) — the array itself.

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

## Output Format
Print the $$$k$$$-th smallest value obtained over all subarrays of length at least $$$2$$$.

## Examples
```input
4
5 2
1 2 3 4 5
2 1
4 3
4 6
1 2 4 8
5 9
1 2 3 4 5
```
```output
1
7
12
3
```

## Note
In the first testcase, we have subarrays with their smallest exclusive-or pair as:

$$$[1,2]: 3$$$

$$$[2,3]: 1$$$

$$$[3,4]: 7$$$

$$$[4,5]: 1$$$

$$$[1,2,3]: 1$$$

$$$[2,3,4]: 1$$$

$$$[3,4,5]: 1$$$

$$$[1,2,3,4]: 1$$$

$$$[2,3,4,5]: 1$$$

$$$[1,2,3,4,5]: 1$$$

The sorted order would be: $$$1, 1, 1, 1, 1, 1, 1, 1, 3, 7$$$. Therefore, the second smallest element would be $$$1$$$.

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: 512.0 MB

# Problem
Monocarp's current password on Codeforces is a string $$$s$$$, consisting of lowercase Latin letters. Monocarp thinks that his current password is too weak, so he wants to insert exactly one lowercase Latin letter into the password to make it stronger. Monocarp can choose any letter and insert it anywhere, even before the first character or after the last character.

Monocarp thinks that the password's strength is proportional to the time it takes him to type the password. The time it takes to type the password is calculated as follows:

- the time to type the first character is $$$2$$$ seconds;
- for each character other than the first, the time it takes to type it is $$$1$$$ second if it is the same as the previous character, or $$$2$$$ seconds otherwise.

For example, the time it takes to type the password abacaba is $$$14$$$; the time it takes to type the password a is $$$2$$$; the time it takes to type the password aaabacc is $$$11$$$.

You have to help Monocarp — insert a lowercase Latin letter into his password so that the resulting password takes the maximum possible amount of time to type.

## 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 $$$s$$$ ($$$1 \le |s| \le 10$$$), consisting of lowercase Latin letters.

## Output Format
For each test case, print one line containing the new password — a string which can be obtained from $$$s$$$ by inserting one lowercase Latin letter. The string you print should have the maximum possible required time to type it. If there are multiple answers, print any of them.

## Examples
```input
4
a
aaa
abb
password
```
```output
wa
aada
abcb
pastsword
```

Now solve the problem and return the code.