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 a permutation$$$^{\dagger}$$$ $$$p$$$ of length $$$n$$$.

We call index $$$x$$$ good if for all $$$y < x$$$ it holds that $$$p_y < p_x$$$ and for all $$$y > x$$$ it holds that $$$p_y > p_x$$$. We call $$$f(p)$$$ the number of good indices in $$$p$$$.

You can perform the following operation: pick $$$2$$$ distinct indices $$$i$$$ and $$$j$$$ and swap elements $$$p_i$$$ and $$$p_j$$$.

Find the maximum value of $$$f(p)$$$ after applying the aforementioned operation exactly once.

$$$^{\dagger}$$$A permutation of length $$$n$$$ is an array consisting of $$$n$$$ distinct integers from $$$1$$$ to $$$n$$$ in arbitrary order. For example, $$$[2,3,1,5,4]$$$ is a permutation, but $$$[1,2,2]$$$ is not a permutation ($$$2$$$ appears twice in the array), and $$$[1,3,4]$$$ is also not a permutation ($$$n=3$$$ but there is $$$4$$$ in the array).

## Input Format
Each test consists of multiple test cases. The first line of contains a single integer $$$t$$$ ($$$1 \le t \le 2 \cdot 10^4$$$) — the number of test cases. The description of the test cases follows.

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

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

It is guaranteed that 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 value of $$$f(p)$$$ after performing the operation exactly once.

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

## Note
In the first test case, $$$p = [1,2,3,4,5]$$$ and $$$f(p)=5$$$ which is already maximum possible. But must perform the operation anyway. We can get $$$f(p)=3$$$ by choosing $$$i=1$$$ and $$$j=2$$$ which makes $$$p = [2,1,3,4,5]$$$.

In the second test case, we can transform $$$p$$$ into $$$[1,2,3,4,5]$$$ by choosing $$$i=1$$$ and $$$j=2$$$. Thus $$$f(p)=5$$$.

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
Berland is very concerned with privacy, so almost all plans and blueprints are secret. However, a spy of the neighboring state managed to steal the Bertown subway scheme.

The Bertown Subway has n stations, numbered from 1 to n, and m bidirectional tunnels connecting them. All Bertown Subway consists of lines. To be more precise, there are two types of lines: circular and radial.

A radial line is a sequence of stations v1, ..., vk (k > 1), where stations vi and vi + 1 (i < k) are connected by a tunnel and no station occurs in the line more than once (vi ≠ vj for i ≠ j).

A loop line is a series of stations, v1, ..., vk (k > 2), where stations vi и vi + 1 are connected by a tunnel. In addition, stations v1 and vk are also connected by a tunnel. No station is occurs in the loop line more than once.

Note that a single station can be passed by any number of lines.

According to Berland standards, there can't be more than one tunnel between two stations and each tunnel belongs to exactly one line. Naturally, each line has at least one tunnel. Between any two stations there is the way along the subway tunnels. In addition, in terms of graph theory, a subway is a vertex cactus: if we consider the subway as a graph in which the stations are the vertexes and the edges are tunnels, then each vertex lies on no more than one simple cycle.

Unfortunately, scheme, stolen by the spy, had only the stations and the tunnels. It was impossible to determine to which line every tunnel corresponds. But to sabotage successfully, the spy needs to know what minimum and maximum number of lines may be in the Bertown subway.

Help him!

## Input Format
The first line contains two integers n and m (1 ≤ n ≤ 105, 0 ≤ m ≤ 3·105) — the number of stations and the number of tunnels, correspondingly.

Each of the next m lines contain two integers — the numbers of stations connected by the corresponding tunnel. The stations are numbered with integers from 1 to n.

It is guaranteed that the graph that corresponds to the subway has no multiple edges or loops, it is connected and it is a vertex cactus.

## Output Format
Print two numbers — the minimum and maximum number of lines correspondingly.

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

## Note
The subway scheme with minimum possible number of lines for the second sample is:

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
Alice and Bob are bored, so they decide to play a game with their wallets. Alice has $$$a$$$ coins in her wallet, while Bob has $$$b$$$ coins in his wallet.

Both players take turns playing, with Alice making the first move. In each turn, the player will perform the following steps in order:

1. Choose to exchange wallets with their opponent, or to keep their current wallets.
2. Remove $$$1$$$ coin from the player's current wallet. The current wallet cannot have $$$0$$$ coins before performing this step.

The player who cannot make a valid move on their turn loses. If both Alice and Bob play optimally, determine who will win the game.

## Input Format
Each test contains multiple test cases. The first line contains a single integer $$$t$$$ ($$$1 \leq t \leq 1000$$$) — the number of test cases. The description of the test cases follows.

The first and only line of each test case contains two integers $$$a$$$ and $$$b$$$ ($$$1 \le a, b \le 10^9$$$) — the number of coins in Alice's and Bob's wallets, respectively.

## Output Format
For each test case, output "Alice" if Alice will win the game, and "Bob" if Bob will win the game.

## Examples
```input
10
1 1
1 4
5 3
4 5
11 9
83 91
1032 9307
839204 7281
1000000000 1000000000
53110 2024
```
```output
Bob
Alice
Bob
Alice
Bob
Bob
Alice
Alice
Bob
Bob
```

## Note
In the first test case, an example of the game is shown below:

- Alice chooses to not swap wallets with Bob in step 1 of her move. Now, $$$a=0$$$ and $$$b=1$$$.
- Since Alice's wallet is empty, Bob must choose to not swap their wallets in step 1 of his move. Now, $$$a=0$$$ and $$$b=0$$$.
- Since both Alice's and Bob's wallets are empty, Alice is unable to make a move. Hence, Bob wins.

In the second test case, an example of the game is shown below:

- Alice chooses to swap wallets with Bob in step 1 of her move. Now, $$$a=3$$$ and $$$b=1$$$.
- Bob chooses to swap wallets with Alice in step 1 of his move. Now, $$$a=1$$$ and $$$b=2$$$.
- Alice chooses to not swap wallets with Bob in step 1 of her move. Now, $$$a=0$$$ and $$$b=2$$$.
- Since Alice's wallet is empty, Bob can only choose to not swap wallets with Alice in step 1 of his move. Now, $$$a=0$$$ and $$$b=1$$$.
- Since Alice's wallet is empty, Alice can only choose to swap wallets with Bob in step 1 of her move. Now, $$$a=0$$$ and $$$b=0$$$.
- Since both Alice's wallet and Bob's wallet are empty, Bob is unable to make a move. Hence, Alice wins.

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 string $$$s$$$ of length $$$n$$$ consisting of characters "+" and "-". $$$s$$$ represents an array $$$a$$$ of length $$$n$$$ defined by $$$a_i=1$$$ if $$$s_i=$$$ "+" and $$$a_i=-1$$$ if $$$s_i=$$$ "-".

You will do the following process to calculate your penalty:

1. Split $$$a$$$ into non-empty arrays $$$b_1,b_2,\ldots,b_k$$$ such that $$$b_1+b_2+\ldots+b_k=a^\dagger$$$, where $$$+$$$ denotes array concatenation.
2. The penalty of a single array is the absolute value of its sum multiplied by its length. In other words, for some array $$$c$$$ of length $$$m$$$, its penalty is calculated as $$$p(c)=|c_1+c_2+\ldots+c_m| \cdot m$$$.
3. The total penalty that you will receive is $$$p(b_1)+p(b_2)+\ldots+p(b_k)$$$.

If you perform the above process optimally, find the minimum possible penalty you will receive.

$$$^\dagger$$$ Some valid ways to split $$$a=[3,1,4,1,5]$$$ into $$$(b_1,b_2,\ldots,b_k)$$$ are $$$([3],[1],[4],[1],[5])$$$, $$$([3,1],[4,1,5])$$$ and $$$([3,1,4,1,5])$$$ while some invalid ways to split $$$a$$$ are $$$([3,1],[1,5])$$$, $$$([3],[\,],[1,4],[1,5])$$$ and $$$([3,4],[5,1,1])$$$.

## Input Format
Each test contains multiple test cases. The first line contains a single integer $$$t$$$ ($$$1 \leq t \leq 1000$$$) — 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 length of string $$$s$$$.

The second line of each test case contains string $$$s$$$ ($$$s_i \in \{ \mathtt{+}, \mathtt{-} \}$$$, $$$|s| = n$$$).

Note that there are no constraints on the sum of $$$n$$$ over all test cases.

## Output Format
For each test case, output a single integer representing the minimum possible penalty you will receive.

## Examples
```input
5
1
+
5
-----
6
+-+-+-
10
--+++++++-
20
+---++++-+++++---++-
```
```output
1
5
0
4
4
```

## Note
In the first test case, we have $$$a=[1]$$$. We can split array $$$a$$$ into $$$([1])$$$. Then, the sum of penalties of the subarrays is $$$p([1]) = 1$$$.

In the second test case, we have $$$a=[-1,-1,-1,-1,-1]$$$. We can split array $$$a$$$ into $$$([-1],[-1],[-1],[-1],[-1])$$$. Then, the sum of penalties of the subarrays is $$$p([-1]) + p([-1]) + p([-1]) + p([-1]) + p([-1]) = 1 + 1 + 1 + 1 + 1 = 5$$$.

In the third test case, we have $$$a=[1,-1,1,-1,1,-1]$$$. We can split array $$$a$$$ into $$$([1,-1,1,-1],[1,-1])$$$. Then, the sum of penalties of the subarrays is $$$p([1,-1,1,-1]) + p([1,-1]) = 0 + 0 = 0$$$.

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: 1.0 seconds
Memory limit: 256.0 MB

# Problem
You are given an array $$$a$$$ of size $$$n$$$. You will do the following process to calculate your penalty:

1. Split array $$$a$$$ into two (possibly empty) subsequences$$$^\dagger$$$ $$$s$$$ and $$$t$$$ such that every element of $$$a$$$ is either in $$$s$$$ or $$$t^\ddagger$$$.
2. For an array $$$b$$$ of size $$$m$$$, define the penalty $$$p(b)$$$ of an array $$$b$$$ as the number of indices $$$i$$$ between $$$1$$$ and $$$m - 1$$$ where $$$b_i < b_{i + 1}$$$.
3. The total penalty you will receive is $$$p(s) + p(t)$$$.

If you perform the above process optimally, find the minimum possible penalty you will receive.

$$$^\dagger$$$ A sequence $$$x$$$ is a subsequence of a sequence $$$y$$$ if $$$x$$$ can be obtained from $$$y$$$ by the deletion of several (possibly, zero or all) elements.

$$$^\ddagger$$$ Some valid ways to split array $$$a=[3,1,4,1,5]$$$ into $$$(s,t)$$$ are $$$([3,4,1,5],[1])$$$, $$$([1,1],[3,4,5])$$$ and $$$([\,],[3,1,4,1,5])$$$ while some invalid ways to split $$$a$$$ are $$$([3,4,5],[1])$$$, $$$([3,1,4,1],[1,5])$$$ and $$$([1,3,4],[5,1])$$$.

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

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

The second line 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 $$$2\cdot 10^5$$$.

## Output Format
For each test case, output a single integer representing the minimum possible penalty you will receive.

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

## Note
In the first test case, a possible way to split $$$a$$$ is $$$s=[2,4,5]$$$ and $$$t=[1,3]$$$. The penalty is $$$p(s)+p(t)=2 + 1 =3$$$.

In the second test case, a possible way to split $$$a$$$ is $$$s=[8,3,1]$$$ and $$$t=[2,1,7,4,3]$$$. The penalty is $$$p(s)+p(t)=0 + 1 =1$$$.

In the third test case, a possible way to split $$$a$$$ is $$$s=[\,]$$$ and $$$t=[3,3,3,3,3]$$$. The penalty is $$$p(s)+p(t)=0 + 0 =0$$$.

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: 1.0 seconds
Memory limit: 256.0 MB

# Problem
There is an edge-weighted complete binary tree with $$$n$$$ leaves. A complete binary tree is defined as a tree where every non-leaf vertex has exactly 2 children. For each non-leaf vertex, we label one of its children as the left child and the other as the right child.

The binary tree has a very strange property. For every non-leaf vertex, one of the edges to its children has weight $$$0$$$ while the other edge has weight $$$1$$$. Note that the edge with weight $$$0$$$ can be connected to either its left or right child.

You forgot what the tree looks like, but luckily, you still remember some information about the leaves in the form of an array $$$a$$$ of size $$$n$$$. For each $$$i$$$ from $$$1$$$ to $$$n$$$, $$$a_i$$$ represents the distance$$$^\dagger$$$ from the root to the $$$i$$$-th leaf in dfs order$$$^\ddagger$$$. Determine whether there exists a complete binary tree which satisfies array $$$a$$$. Note that you do not need to reconstruct the tree.

$$$^\dagger$$$ The distance from vertex $$$u$$$ to vertex $$$v$$$ is defined as the sum of weights of the edges on the path from vertex $$$u$$$ to vertex $$$v$$$.

$$$^\ddagger$$$ The dfs order of the leaves is found by calling the following $$$\texttt{dfs}$$$ function on the root of the binary tree.

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

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

The second line of each test case contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$0 \le a_i \le n - 1$$$) — the distance from the root to the $$$i$$$-th leaf.

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, print "YES" if there exists a complete binary tree which satisfies array $$$a$$$ and "NO" otherwise.

You may print each letter in any case (for example, "YES", "Yes", "yes", "yEs" will all be recognized as a positive answer).

## Examples
```input
2
5
2 1 0 1 1
5
1 0 2 1 3
```
```output
YES
NO
```

## Note
In the first test case, the following tree satisfies the array.

In the second test case, it can be proven that there is no complete binary tree that satisfies the array.

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
Bob has a rectangle of size $$$a \times b$$$. He tries to cut this rectangle into two rectangles with integer sides by making a cut parallel to one of the sides of the original rectangle. Then Bob tries to form some other rectangle from the two resulting rectangles, and he can rotate and move these two rectangles as he wishes.

Note that if two rectangles differ only by a $$$90^{\circ}$$$ rotation, they are considered the same. For example, the rectangles $$$6 \times 4$$$ and $$$4 \times 6$$$ are considered the same.

Thus, from the $$$2 \times 6$$$ rectangle, another rectangle can be formed, because it can be cut into two $$$2 \times 3$$$ rectangles, and then these two rectangles can be used to form the $$$4 \times 3$$$ rectangle, which is different from the $$$2 \times 6$$$ rectangle.

However, from the $$$2 \times 1$$$ rectangle, another rectangle cannot be formed, because it can only be cut into two rectangles of $$$1 \times 1$$$, and from these, only the $$$1 \times 2$$$ and $$$2 \times 1$$$ rectangles can be formed, which are considered the same.

Help Bob determine if he can obtain some other rectangle, or if he is just wasting his time.

## Input Format
Each test consists of multiple test cases. The first line contains a single integer $$$t$$$ ($$$1 \leq t \leq 10^4$$$) — the number of test cases. This is followed by the description of the test cases.

The single line of each test case contains two integers $$$a$$$ and $$$b$$$ ($$$1 \le a, b \le 10^9$$$) — the size of Bob's rectangle.

## Output Format
For each test case, output "Yes" if Bob can obtain another rectangle from the $$$a \times b$$$ rectangle. Otherwise, output "No".

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 answers.

## Examples
```input
7
1 1
2 1
2 6
3 2
2 2
2 4
6 3
```
```output
No
No
Yes
Yes
Yes
Yes
No
```

## Note
In the first test case, the $$$1 \times 1$$$ rectangle cannot be cut into two rectangles, so another rectangle cannot be obtained from it.

In the fourth test case, the $$$3 \times 2$$$ rectangle can be cut into two $$$3 \times 1$$$ rectangles, and from these, the $$$1 \times 6$$$ rectangle can be formed.

In the fifth test case, the $$$2 \times 2$$$ rectangle can be cut into two $$$1 \times 2$$$ rectangles, and from these, the $$$1 \times 4$$$ rectangle can be formed.

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
In a well-known school, a physical education lesson took place. As usual, everyone was lined up and asked to settle in "the first–$$$k$$$-th" position.

As is known, settling in "the first–$$$k$$$-th" position occurs as follows: the first $$$k$$$ people have numbers $$$1, 2, 3, \ldots, k$$$, the next $$$k - 2$$$ people have numbers $$$k - 1, k - 2, \ldots, 2$$$, the next $$$k$$$ people have numbers $$$1, 2, 3, \ldots, k$$$, and so on. Thus, the settling repeats every $$$2k - 2$$$ positions. Examples of settling are given in the "Note" section.

The boy Vasya constantly forgets everything. For example, he forgot the number $$$k$$$ described above. But he remembers the position he occupied in the line, as well as the number he received during the settling. Help Vasya understand how many natural numbers $$$k$$$ fit under the given constraints.

Note that the settling exists if and only if $$$k > 1$$$. In particular, this means that the settling does not exist for $$$k = 1$$$.

## Input Format
Each test consists of multiple test cases. The first line contains a single integer $$$t$$$ ($$$1 \leq t \leq 100$$$) — the number of test cases. This is followed by the description of the test cases.

The only line of each test case contains two integers $$$n$$$ and $$$x$$$ ($$$1 \le x < n \le 10^9$$$) — Vasya's position in the line and the number Vasya received during the settling.

## Output Format
For each test case, output a single integer — the number of different $$$k$$$ that fit under the given constraints.

It can be proven that under the given constraints, the answer is finite.

## Examples
```input
5
10 2
3 1
76 4
100 99
1000000000 500000000
```
```output
4
1
9
0
1
```

## Note
In the first test case, $$$k$$$ equals $$$2, 3, 5, 6$$$ are suitable.

An example of settling for these $$$k$$$:

$$$k$$$ / №$$$1$$$$$$2$$$$$$3$$$$$$4$$$$$$5$$$$$$6$$$$$$7$$$$$$8$$$$$$9$$$$$$10$$$$$$2$$$$$$1$$$$$$2$$$$$$1$$$$$$2$$$$$$1$$$$$$2$$$$$$1$$$$$$2$$$$$$1$$$$$$2$$$$$$3$$$$$$1$$$$$$2$$$$$$3$$$$$$2$$$$$$1$$$$$$2$$$$$$3$$$$$$2$$$$$$1$$$$$$2$$$$$$5$$$$$$1$$$$$$2$$$$$$3$$$$$$4$$$$$$5$$$$$$4$$$$$$3$$$$$$2$$$$$$1$$$$$$2$$$$$$6$$$$$$1$$$$$$2$$$$$$3$$$$$$4$$$$$$5$$$$$$6$$$$$$5$$$$$$4$$$$$$3$$$$$$2$$$

In the second test case, $$$k = 2$$$ is suitable.

Now solve the problem and return the code.