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
Given two arrays $$$a$$$ and $$$b$$$, both of length $$$n$$$. Elements of both arrays indexed from $$$1$$$ to $$$n$$$. You are constructing a directed graph, where edge from $$$u$$$ to $$$v$$$ ($$$u\neq v$$$) exists if $$$a_u-a_v \ge b_u-b_v$$$.

A vertex $$$V$$$ is called strong if there exists a path from $$$V$$$ to all other vertices.

A path in a directed graph is a chain of several vertices, connected by edges, such that moving from the vertex $$$u$$$, along the directions of the edges, the vertex $$$v$$$ can be reached.

Your task is to find all strong vertices.

For example, if $$$a=[3,1,2,4]$$$ and $$$b=[4,3,2,1]$$$, the graph will look like this:

The graph has only one strong vertex with number $$$4$$$

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

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

The second line of each test case contains $$$n$$$ integers $$$a_1,a_2 \dots a_n$$$ ($$$-10^9 \le a_i \le 10^9$$$) — the array $$$a$$$.

The third line of each test case contains $$$n$$$ integers $$$b_1,b_2 \dots b_n$$$ ($$$-10^9 \le b_i \le 10^9$$$) — the array $$$b$$$.

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

## Output Format
For each test case, output two lines: in the first line, output the number of strong vertices, and in the second line, output all strong vertices in ascending order.

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

## Note
The first sample is covered in the problem statement.

For the second sample, the graph looks like this:

The graph has two strong vertices with numbers $$$3$$$ and $$$5$$$. Note that there is a bidirectional edge between vertices $$$3$$$ and $$$5$$$.

In the third sample, the vertices are connected by a single directed edge from vertex $$$2$$$ to vertex $$$1$$$, so the only strong vertex is $$$2$$$.

In the fourth sample, all vertices are connected to each other by bidirectional edges, so there is a path from every vertex to any other vertex.

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 $$$n$$$ points with integer coordinates $$$x_1,\dots x_n$$$, which lie on a number line.

For some integer $$$s$$$, we construct segments [$$$s,x_1$$$], [$$$s,x_2$$$], $$$\dots$$$, [$$$s,x_n$$$]. Note that if $$$x_i<s$$$, then the segment will look like [$$$x_i,s$$$]. The segment [$$$a, b$$$] covers all integer points $$$a, a+1, a+2, \dots, b$$$.

We define the power of a point $$$p$$$ as the number of segments that intersect the point with coordinate $$$p$$$, denoted as $$$f_p$$$.

Your task is to compute $$$\sum\limits_{p=1}^{10^9}f_p$$$ for each $$$s \in \{x_1,\dots,x_n\}$$$, i.e., the sum of $$$f_p$$$ for all integer points from $$$1$$$ to $$$10^9$$$.

For example, if the initial coordinates are $$$[1,2,5,7,1]$$$ and we choose $$$s=5$$$, then the segments will be: $$$[1,5]$$$,$$$[2,5]$$$,$$$[5,5]$$$,$$$[5,7]$$$,$$$[1,5]$$$. And the powers of the points will be: $$$f_1=2, f_2=3, f_3=3, f_4=3, f_5=5, f_6=1, f_7=1, f_8=0, \dots, f_{10^9}=0$$$. Their sum is $$$2+3+3+3+5+1+1=18$$$.

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

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

The second line contains $$$n$$$ integers $$$x_1,x_2 \dots x_n$$$ ($$$1 \le x_i \le 10^9$$$) — the coordinates of the points.

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

## Output Format
For each test case, output $$$n$$$ integers, where the $$$i$$$-th integer is equal to the sum of the powers of all points for $$$s=x_i$$$.

## Examples
```input
3
3
1 4 3
5
1 2 5 7 1
4
1 10 100 1000
```
```output
8 7 6
16 15 18 24 16
1111 1093 1093 2893
```

## Note
In the first test case we first choose $$$s=x_1=1$$$, then the following segments are formed: $$$[1,1]$$$,$$$[1,4]$$$,$$$[1,3]$$$.

The powers of the points will be as follows: $$$f_1=3, f_2=2, f_3=2, f_4=1, f_5=0 \dots$$$ The sum of powers of the points: $$$3+2+2+1+0+\dots+0=8$$$.

After that we choose $$$s=x_2=4$$$. Then there will be such segments: $$$[1,4]$$$,$$$[4,4]$$$,$$$[3,4]$$$, and powers of the points are $$$f_1=1, f_2=1, f_3=2, f_4=3$$$.

At the end we take $$$s=x_3=3$$$ and the segments look like this: $$$[1,3]$$$,$$$[3,4]$$$,$$$[3,3]$$$, the powers of the points are $$$f_1=1, f_2=1, f_3=3, f_4=1$$$.

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: 4.0 seconds
Memory limit: 256.0 MB

# Problem
You have an array $$$a$$$ of length $$$n$$$.

Your task is to answer $$$q$$$ queries: given $$$x,y$$$, find the number of pairs $$$i$$$ and $$$j$$$ ($$$1 \le i < j \le n$$$) that both $$$a_i + a_j = x$$$ and $$$a_i \cdot a_j = y$$$.

That is, for the array $$$[1,3,2]$$$ and asking for $$$x=3,y=2$$$ the answer is $$$1$$$:

- $$$i=1$$$ and $$$j=2$$$ fail because $$$1 + 3 = 4$$$ and not $$$3,$$$ also $$$1 \cdot 3=3$$$ and not $$$2$$$;
- $$$i=1$$$ and $$$j=3$$$ satisfies both conditions;
- $$$i=2$$$ and $$$j=3$$$ fail because $$$3 + 2 = 5$$$ and not $$$3,$$$ also $$$3 \cdot 2=6$$$ and not $$$2$$$;

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

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

The third line of each test case contains $$$n$$$ integers $$$a_1,a_2,\dots,a_n$$$ ($$$1 \le |a_i| \le 10^9$$$) — array $$$a$$$.

The fourth line of each test case contains the integer $$$q$$$ ($$$1 \le q \le 2\cdot 10^5$$$) — the number of requests.

The next $$$q$$$ lines contain two numbers each $$$x$$$ and $$$y$$$ ($$$1 \le |x|\le 2\cdot 10^9,1\le |y|\le 10^{18}$$$) — request.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2\cdot 10^5$$$. This is also guaranteed for the sum of $$$q$$$ values.

## Output Format
For each test case print a line with $$$q$$$ numbers — the answers to the queries.

## Examples
```input
3
3
1 3 2
4
3 2
5 6
3 1
5 5
4
1 1 1 1
1
2 1
6
1 4 -2 3 3 3
3
2 -8
-1 -2
7 12
```
```output
1 1 0 0 
6 
1 1 3
```

## Note
For the first test case, let's analyze each pair of numbers separately:

- pair $$$(a_1,a_2)$$$: $$$a_1 + a_2 = 4$$$, $$$a_1 \cdot a_2 = 3$$$
- pair $$$(a_1,a_3)$$$: $$$a_1 + a_3 = 3$$$, $$$a_1 \cdot a_3 = 2$$$
- pair $$$(a_2,a_3)$$$: $$$a_2 + a_3 = 5$$$, $$$a_2 \cdot a_3 = 6$$$

In the second test case, all combinations of pairs are suitable.

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 $$$n\times m$$$ board divided into cells. There are also some dominoes on this board. Each domino covers two adjacent cells (that is, two cells that share a side), and no two dominoes overlap.

Piet thinks that this board is too boring and it needs to be painted. He will paint the cells of the dominoes black and white. He calls the painting beautiful if all of the following conditions hold:

- for each domino, one of its cells is painted white and the other is painted black;
- for each row, the number of black cells in this row equals the number of white cells in this row;
- for each column, the number of black cells in this column equals the number of white cells in this column.

Note that the cells that are not covered by dominoes are not painted at all, they are counted as neither black nor white.

Help Piet produce a beautiful painting or tell that it is impossible.

## Input Format
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 10\,000$$$). The description of the test cases follows.

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$2\le n, m\le 500$$$).

The next $$$n$$$ lines describe the tiling of the board, row by row from top to bottom. Each of these lines contains $$$m$$$ characters, describing the cells in the corresponding row from left to right. Each character is one of U, D, L, R, or ., meaning that the cell is covered with a top, bottom, left, right half of a domino or nothing, respectively. The tiling is guaranteed to be valid.

It is guaranteed that the sum of $$$n \cdot m$$$ over all test cases does not exceed $$$250\,000$$$.

## Output Format
For each test case, output a single integer $$$-1$$$ if a beautiful painting does not exist. Otherwise, print $$$n$$$ lines, each containing $$$m$$$ characters, describing the colors in the corresponding row of a beautiful painting. Every character corresponding to a cell not covered by a domino must be . (a dot), and all other characters must be B if the corresponding cell is black or W if it is white.

If there are multiple solutions, print any of them.

## Examples
```input
3
4 6
..LR..
ULRU..
DLRDUU
..LRDD
5 4
.LR.
.UU.
UDDU
D..D
LR..
2 2
..
..
```
```output
..WB..
WWBB..
BBWWWB
..BWBW
-1
..
..
```

## Note
In the first test case, the answer is illustrated below:

In the second test case, it is impossible to paint all cells the right way.

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 an array of integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le n$$$). You can perform the following operation several (possibly, zero) times:

- pick an arbitrary $$$i$$$ and perform swap$$$(a_i, a_{a_i})$$$.

How many distinct arrays is it possible to attain? Output the answer modulo $$$(10^9 + 7)$$$.

## Input Format
The first line contains an integer $$$n$$$ ($$$1 \le n \le 10^6$$$).

The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1\le a_i\le n$$$).

## Output Format
Output the number of attainable arrays modulo $$$(10^9 + 7)$$$.

## Examples
```input
3
1 1 2
```
```output
2
```
-----
```input
4
2 1 4 3
```
```output
4
```
-----
```input
6
2 3 1 1 1 2
```
```output
18
```

## Note
In the first example, the initial array is $$$[1, 1, 2]$$$. If we perform the operation with $$$i = 3$$$, we swap $$$a_3$$$ and $$$a_2$$$, obtaining $$$[1, 2, 1]$$$. One can show that there are no other attainable arrays.

In the second example, the four attainable arrays are $$$[2, 1, 4, 3]$$$, $$$[1, 2, 4, 3]$$$, $$$[1, 2, 3, 4]$$$, $$$[2, 1, 3, 4]$$$. One can show that there are no other attainable arrays.

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: 5.0 seconds
Memory limit: 1024.0 MB

# Problem
You are given a tree with $$$n$$$ vertices labeled $$$1, 2, \ldots, n$$$. The length of a simple path in the tree is the number of vertices in it.

You are to select a set of simple paths of length at least $$$2$$$ each, but you cannot simultaneously select two distinct paths contained one in another. Find the largest possible size of such a set.

Formally, a set $$$S$$$ of vertices is called a route if it contains at least two vertices and coincides with the set of vertices of a simple path in the tree. A collection of distinct routes is called a timetable. A route $$$S$$$ in a timetable $$$T$$$ is called redundant if there is a different route $$$S' \in T$$$ such that $$$S \subset S'$$$. A timetable is called efficient if it contains no redundant routes. Find the largest possible number of routes in an efficient timetable.

## Input Format
The first line contains a single integer $$$n$$$ ($$$2 \le n \le 3000$$$).

The $$$i$$$-th of the following $$$n - 1$$$ lines contains two integers $$$u_i$$$ and $$$v_i$$$ ($$$1 \le u_i, v_i \le n$$$, $$$u_i \neq v_i$$$) — the numbers of vertices connected by the $$$i$$$-th edge.

It is guaranteed that the given edges form a tree.

## Output Format
Print a single integer — the answer to the problem.

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

## Note
In the first example, possible efficient timetables are $$$\{\{1, 2\}, \{1, 3\}, \{1, 4\}\}$$$ and $$$\{\{1, 2, 3\}, \{1, 2, 4\}, \{1, 3, 4\}\}$$$.

In the second example, we can choose $$$\{ \{1, 2, 3\}, \{2, 3, 4\}, \{3, 4, 6\}, \{2, 3, 5\}, \{3, 4, 5\}, \{3, 4, 7\}, \{4, 6, 7\}\}$$$.

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$$$^\dagger$$$ $$$a$$$ of size $$$n$$$. You can do the following operation

- Select an index $$$i$$$ from $$$2$$$ to $$$n - 1$$$ such that $$$a_{i - 1} < a_i$$$ and $$$a_i > a_{i+1}$$$. Swap $$$a_i$$$ and $$$a_{i+1}$$$.

Determine whether it is possible to sort the permutation after a finite number of operations.

$$$^\dagger$$$ A permutation 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 contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 5000$$$). Description of the test cases follows.

The first line of each test case contains a single integer $$$n$$$ ($$$3 \le n \le 10$$$) — the size of the permutation.

The second line of each test case contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$1 \le a_i \le n$$$) — the elements of permutation $$$a$$$.

## Output Format
For each test case, print "YES" if it is possible to sort the permutation, and "NO" otherwise.

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

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

## Note
In the first test case, the permutation is already sorted.

In the second test case, we can choose index $$$i=2$$$ as $$$1<3$$$ and $$$3>2$$$ to form $$$[1, 2, 3, 5, 4]$$$. Then, we can choose index $$$i=4$$$ as $$$3<5$$$ and $$$5>4$$$ to form $$$[1, 2, 3, 4, 5]$$$.

In the third test case, it can be proven that it is impossible to sort the permutation.

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 $$$\texttt{A}$$$ and $$$\texttt{B}$$$. You are allowed to do the following operation:

- Choose an index $$$1 \le i \le n - 1$$$ such that $$$s_i = \texttt{A}$$$ and $$$s_{i + 1} = \texttt{B}$$$. Then, swap $$$s_i$$$ and $$$s_{i+1}$$$.

You are only allowed to do the operation at most once for each index $$$1 \le i \le n - 1$$$. However, you can do it in any order you want. Find the maximum number of operations that you can carry out.

## Input Format
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 1000$$$). 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 string $$$s$$$.

The second line of each test case contains the string $$$s$$$ ($$$s_i=\texttt{A}$$$ or $$$s_i=\texttt{B}$$$).

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 a single integer containing the maximum number of operations that you can carry out.

## Examples
```input
3
2
AB
4
BBBA
4
AABB
```
```output
1
0
3
```

## Note
In the first test case, we can do the operation exactly once for $$$i=1$$$ as $$$s_1=\texttt{A}$$$ and $$$s_2=\texttt{B}$$$.

In the second test case, it can be proven that it is not possible to do an operation.

In the third test case, we can do an operation on $$$i=2$$$ to form $$$\texttt{ABAB}$$$, then another operation on $$$i=3$$$ to form $$$\texttt{ABBA}$$$, and finally another operation on $$$i=1$$$ to form $$$\texttt{BABA}$$$. Note that even though at the end, $$$s_2 = \texttt{A}$$$ and $$$s_3 = \texttt{B}$$$, we cannot do an operation on $$$i=2$$$ again as we can only do the operation at most once for each index.

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 $$$1$$$-indexed array $$$a$$$ of length $$$n$$$ where each element is $$$1$$$ or $$$2$$$.

Process $$$q$$$ queries of the following two types:

- "1 s": check if there exists a subarray$$$^{\dagger}$$$ of $$$a$$$ whose sum equals to $$$s$$$.
- "2 i v": change $$$a_i$$$ to $$$v$$$.

$$$^{\dagger}$$$ An array $$$b$$$ is a subarray of an array $$$a$$$ if $$$b$$$ can be obtained from $$$a$$$ by deletion of several (possibly, zero or all) elements from the beginning and several (possibly, zero or all) elements from the end. In particular, an array is a subarray of itself.

## Input Format
Each test contains multiple test cases. The first line 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 $$$q$$$ ($$$1\le n,q\le 10^5$$$) — the length of array $$$a$$$ and the number of queries.

The second line of each test case contains $$$n$$$ integers $$$a_1,a_2,\ldots,a_n$$$ ($$$a_i$$$ is $$$1$$$ or $$$2$$$) — the elements of array $$$a$$$.

Each of the following $$$q$$$ lines of each test case contains some integers. The first integer $$$\mathrm{op}$$$ is either $$$1$$$ or $$$2$$$.

- If $$$\mathrm{op}$$$ is $$$1$$$, it is followed by one integer $$$s$$$ ($$$1 \leq s \leq 2n$$$).
- If $$$\mathrm{op}$$$ is $$$2$$$, it is followed by two integers $$$i$$$ and $$$v$$$ ($$$1 \leq i \leq n$$$, $$$v$$$ is $$$1$$$ or $$$2$$$).

It is guaranteed that the sum of $$$n$$$ and the sum of $$$q$$$ over all test cases both do not exceed $$$10^5$$$.

## Output Format
For each query with $$$\mathrm{op}=1$$$, output "YES" in one line if there exists a subarray of $$$a$$$ whose sum is equals to $$$s$$$, 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 responses.

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

## Note
Consider the first example:

- The answer for the first query is "YES" because $$$a_1+a_2+a_3=2+1+2=5$$$.
- The answer for the second query is "YES" because $$$a_1+a_2+a_3+a_4=2+1+2+1=6$$$.
- The answer for the third query is "NO" because we cannot find any subarray of $$$a$$$ whose sum is $$$7$$$.
- After the fourth query, the array $$$a$$$ becomes $$$[2,1,2,2,2]$$$.
- The answer for the fifth query is "YES" because $$$a_2+a_3+a_4+a_5=1+2+2+2=7$$$.

Now solve the problem and return the code.