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
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
You are given a positive integer $$$n$$$. Let's call some positive integer $$$a$$$ without leading zeroes palindromic if it remains the same after reversing the order of its digits. Find the number of distinct ways to express $$$n$$$ as a sum of positive palindromic integers. Two ways are considered different if the frequency of at least one palindromic integer is different in them. For example, $$$5=4+1$$$ and $$$5=3+1+1$$$ are considered different but $$$5=3+1+1$$$ and $$$5=1+3+1$$$ are considered the same.

Formally, you need to find the number of distinct multisets of positive palindromic integers the sum of which is equal to $$$n$$$.

Since the answer can be quite large, print it modulo $$$10^9+7$$$.

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

Each testcase contains a single line of input containing a single integer $$$n$$$ ($$$1\leq n\leq 4\cdot 10^4$$$) — the required sum of palindromic integers.

## Output Format
For each testcase, print a single integer denoting the required answer modulo $$$10^9+7$$$.

## Examples
```input
2
5
12
```
```output
7
74
```

## Note
For the first testcase, there are $$$7$$$ ways to partition $$$5$$$ as a sum of positive palindromic integers:

- $$$5=1+1+1+1+1$$$
- $$$5=1+1+1+2$$$
- $$$5=1+2+2$$$
- $$$5=1+1+3$$$
- $$$5=2+3$$$
- $$$5=1+4$$$
- $$$5=5$$$

For the second testcase, there are total $$$77$$$ ways to partition $$$12$$$ as a sum of positive integers but among them, the partitions $$$12=2+10$$$, $$$12=1+1+10$$$ and $$$12=12$$$ are not valid partitions of $$$12$$$ as a sum of positive palindromic integers because $$$10$$$ and $$$12$$$ are not palindromic. So, there are $$$74$$$ ways to partition $$$12$$$ as a sum of positive palindromic integers.

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
Tokitsukaze has a sequence with length of $$$n$$$. She likes gems very much. There are $$$n$$$ kinds of gems. The gems of the $$$i$$$-th kind are on the $$$i$$$-th position, and there are $$$a_i$$$ gems of the same kind on that position. Define $$$G(l,r)$$$ as the multiset containing all gems on the segment $$$[l,r]$$$ (inclusive).

A multiset of gems can be represented as $$$S=[s_1,s_2,\ldots,s_n]$$$, which is a non-negative integer sequence of length $$$n$$$ and means that $$$S$$$ contains $$$s_i$$$ gems of the $$$i$$$-th kind in the multiset. A multiset $$$T=[t_1,t_2,\ldots,t_n]$$$ is a multisubset of $$$S=[s_1,s_2,\ldots,s_n]$$$ if and only if $$$t_i\le s_i$$$ for any $$$i$$$ satisfying $$$1\le i\le n$$$.

Now, given two positive integers $$$k$$$ and $$$p$$$, you need to calculate the result of

$$$$$$\sum_{l=1}^n \sum_{r=l}^n\sum\limits_{[t_1,t_2,\cdots,t_n] \subseteq G(l,r)}\left(\left(\sum_{i=1}^n p^{t_i}t_i^k\right)\left(\sum_{i=1}^n[t_i>0]\right)\right),$$$$$$

where $$$[t_i>0]=1$$$ if $$$t_i>0$$$ and $$$[t_i>0]=0$$$ if $$$t_i=0$$$.

Since the answer can be quite large, print it modulo $$$998\,244\,353$$$.

## Input Format
The first line contains three integers $$$n$$$, $$$k$$$ and $$$p$$$ ($$$1\le n \le 10^5$$$; $$$1\le k\le 10^5$$$; $$$2\le p\le 998\,244\,351$$$) — the length of the sequence, the numbers $$$k$$$ and $$$p$$$.

The second line contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1\le a_i\le 998\,244\,351$$$) — the number of gems on the $$$i$$$-th position.

## Output Format
Print a single integers — the result modulo $$$998\,244\,353$$$.

## Examples
```input
5 2 2
1 1 1 2 2
```
```output
6428
```
-----
```input
6 2 2
2 2 2 2 2 3
```
```output
338940
```

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 a permutation $$$p$$$ of integers from $$$0$$$ to $$$n-1$$$ (each of them occurs exactly once). Initially, the permutation is not sorted (that is, $$$p_i>p_{i+1}$$$ for at least one $$$1 \le i \le n - 1$$$).

The permutation is called $$$X$$$-sortable for some non-negative integer $$$X$$$ if it is possible to sort the permutation by performing the operation below some finite number of times:

- Choose two indices $$$i$$$ and $$$j$$$ $$$(1 \le i \lt j \le n)$$$ such that $$$p_i \& p_j = X$$$.
- Swap $$$p_i$$$ and $$$p_j$$$.

Here $$$\&$$$ denotes the bitwise AND operation.

Find the maximum value of $$$X$$$ such that $$$p$$$ is $$$X$$$-sortable. It can be shown that there always exists some value of $$$X$$$ such that $$$p$$$ is $$$X$$$-sortable.

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

The first line of each test case contains a single integer $$$n$$$ $$$(2 \le n \le 2 \cdot 10^5)$$$  — the length of the permutation.

The second line of each test case contains $$$n$$$ integers $$$p_1, p_2, ..., p_n$$$ ($$$0 \le p_i \le n-1$$$, all $$$p_i$$$ are distinct)  — the elements of $$$p$$$. It is guaranteed that $$$p$$$ is not sorted.

It is guaranteed that the sum of $$$n$$$ over all cases does not exceed $$$2 \cdot 10^5$$$.

## Output Format
For each test case output a single integer — the maximum value of $$$X$$$ such that $$$p$$$ is $$$X$$$-sortable.

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

## Note
In the first test case, the only $$$X$$$ for which the permutation is $$$X$$$-sortable are $$$X = 0$$$ and $$$X = 2$$$, maximum of which is $$$2$$$.

Sorting using $$$X = 0$$$:

- Swap $$$p_1$$$ and $$$p_4$$$, $$$p = [2, 1, 3, 0]$$$.
- Swap $$$p_3$$$ and $$$p_4$$$, $$$p = [2, 1, 0, 3]$$$.
- Swap $$$p_1$$$ and $$$p_3$$$, $$$p = [0, 1, 2, 3]$$$.

Sorting using $$$X = 2$$$:

- Swap $$$p_3$$$ and $$$p_4$$$, $$$p = [0, 1, 2, 3]$$$.

In the second test case, we must swap $$$p_1$$$ and $$$p_2$$$ which is possible only with $$$X = 0$$$.

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: 1.0 seconds
Memory limit: 256.0 MB

# Problem
You are given an array $$$a$$$ of $$$n$$$ positive integers.

Let $$$\text{LIS}(a)$$$ denote the length of longest strictly increasing subsequence of $$$a$$$. For example,

- $$$\text{LIS}([2, \underline{1}, 1, \underline{3}])$$$ = $$$2$$$.
- $$$\text{LIS}([\underline{3}, \underline{5}, \underline{10}, \underline{20}])$$$ = $$$4$$$.
- $$$\text{LIS}([3, \underline{1}, \underline{2}, \underline{4}])$$$ = $$$3$$$.

We define array $$$a'$$$ as the array obtained after reversing the array $$$a$$$ i.e. $$$a' = [a_n, a_{n-1}, \ldots , a_1]$$$.

The beauty of array $$$a$$$ is defined as $$$min(\text{LIS}(a),\text{LIS}(a'))$$$.

Your task is to determine the maximum possible beauty of the array $$$a$$$ if you can rearrange the array $$$a$$$ arbitrarily.

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

The first line of each test case contains a single integer $$$n$$$ $$$(1 \leq n \leq 2\cdot 10^5)$$$  — the length of array $$$a$$$.

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

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 integer  — the maximum possible beauty of $$$a$$$ after rearranging its elements arbitrarily.

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

## Note
In the first test case, $$$a$$$ = $$$[6, 6, 6]$$$ and $$$a'$$$ = $$$[6, 6, 6]$$$. $$$\text{LIS}(a) = \text{LIS}(a')$$$ = $$$1$$$. Hence the beauty is $$$min(1, 1) = 1$$$.

In the second test case, $$$a$$$ can be rearranged to $$$[2, 5, 4, 5, 4, 2]$$$. Then $$$a'$$$ = $$$[2, 4, 5, 4, 5, 2]$$$. $$$\text{LIS}(a) = \text{LIS}(a') = 3$$$. Hence the beauty is $$$3$$$ and it can be shown that this is the maximum possible beauty.

In the third test case, $$$a$$$ can be rearranged to $$$[1, 2, 3, 2]$$$. Then $$$a'$$$ = $$$[2, 3, 2, 1]$$$. $$$\text{LIS}(a) = 3$$$, $$$\text{LIS}(a') = 2$$$. Hence the beauty is $$$min(3, 2) = 2$$$ and it can be shown that $$$2$$$ is the maximum possible beauty.

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
For an array $$$[b_1, b_2, \ldots, b_m]$$$ define its number of inversions as the number of pairs $$$(i, j)$$$ of integers such that $$$1 \le i < j \le m$$$ and $$$b_i>b_j$$$. Let's call array $$$b$$$ odd if its number of inversions is odd.

For example, array $$$[4, 2, 7]$$$ is odd, as its number of inversions is $$$1$$$, while array $$$[2, 1, 4, 3]$$$ isn't, as its number of inversions is $$$2$$$.

You are given a permutation $$$[p_1, p_2, \ldots, p_n]$$$ of integers from $$$1$$$ to $$$n$$$ (each of them appears exactly once in the permutation). You want to split it into several consecutive subarrays (maybe just one), so that the number of the odd subarrays among them is as large as possible.

What largest number of these subarrays may be odd?

## Input Format
The first line of the input contains a single integer $$$t$$$ ($$$1 \le t \le 10^5$$$)  — 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 size of the permutation.

The second line of each test case contains $$$n$$$ integers $$$p_1, p_2, \ldots, p_n$$$ ($$$1 \le p_i \le n$$$, all $$$p_i$$$ are distinct)  — the elements of the permutation.

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

## Output Format
For each test case output a single integer  — the largest possible number of odd subarrays that you can get after splitting the permutation into several consecutive subarrays.

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

## Note
In the first and third test cases, no matter how we split our permutation, there won't be any odd subarrays.

In the second test case, we can split our permutation into subarrays $$$[4, 3], [2, 1]$$$, both of which are odd since their numbers of inversions are $$$1$$$.

In the fourth test case, we can split our permutation into a single subarray $$$[2, 1]$$$, which is odd.

In the fifth test case, we can split our permutation into subarrays $$$[4, 5], [6, 1, 2, 3]$$$. The first subarray has $$$0$$$ inversions, and the second has $$$3$$$, so it is odd.

Now solve the problem and return the code.