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: 1.0 seconds
Memory limit: 1024.0 MB

# Problem
You are given a one-based array consisting of $$$N$$$ integers: $$$A_1, A_2, \cdots, A_N$$$. Initially, the value of each element is set to $$$0$$$.

There are $$$M$$$ operations (numbered from $$$1$$$ to $$$M$$$). Operation $$$i$$$ is represented by $$$\langle L_i, R_i, X_i \rangle$$$. If operation $$$i$$$ is executed, all elements $$$A_j$$$ for $$$L_i \leq j \leq R_i$$$ will be increased by $$$X_i$$$.

You have to answer $$$Q$$$ independent queries. Each query is represented by $$$\langle K, S, T \rangle$$$ which represents the following task. Choose a range $$$[l, r]$$$ satisfying $$$S \leq l \leq r \leq T$$$, and execute operations $$$l, l + 1, \dots, r$$$. The answer to the query is the maximum value of $$$A_K$$$ after the operations are executed among all possible choices of $$$l$$$ and $$$r$$$.

## Input Format
The first line consists of two integers $$$N$$$ $$$M$$$ ($$$1 \leq N, M \leq 100\,000$$$).

Each of the next $$$M$$$ lines consists of three integers $$$L_i$$$ $$$R_i$$$ $$$X_i$$$ ($$$1 \leq L_i \leq R_i \leq N; -100\,000 \leq X_i \leq 100\,000$$$).

The following line consists of an integer $$$Q$$$ ($$$1 \leq Q \leq 100\,000$$$).

Each of the next $$$Q$$$ lines consists of three integers $$$K$$$ $$$S$$$ $$$T$$$ ($$$1 \leq K \leq N; 1 \leq S \leq T \leq M$$$).

## Output Format
For each query, output in a single line, an integer which represent the answer of the query.

## Examples
```input
2 6
1 1 -50
1 2 -20
2 2 -30
1 1 60
1 2 40
2 2 10
5
1 1 6
2 1 6
1 1 3
2 1 3
1 1 2
```
```output
100
50
0
0
-20
```
-----
```input
5 3
1 3 3
2 4 -2
3 5 3
6
1 1 3
2 1 3
3 1 3
3 2 3
2 2 3
2 2 2
```
```output
3
3
4
3
0
-2
```

## Note
Explanation for the sample input/output #1

For query $$$1$$$, one of the solutions is to execute operation $$$4$$$ and $$$5$$$.

For query $$$2$$$, one of the solutions is to execute operation $$$4$$$, $$$5$$$, and $$$6$$$.

For query $$$3$$$, the only solution is to execute operation $$$3$$$.

For query $$$4$$$, the only solution is to execute operation $$$1$$$.

For query $$$6$$$, the only solution is to execute operation $$$2$$$.

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: 1.0 seconds
Memory limit: 1024.0 MB

# Problem
You are playing "Grid Game 2" with your friend. There is a grid with $$$10^9$$$ rows (numbered from $$$1$$$ to $$$10^9$$$) and $$$10^9$$$ columns (numbered from $$$1$$$ to $$$10^9$$$). The cell at row $$$r$$$ and column $$$c$$$ is denoted as $$$(r, c)$$$.

Each cell can have a colour of either black or white. Initially, there are exactly $$$N$$$ black cells (numbered from $$$1$$$ to $$$N$$$). Black cell $$$i$$$ is located at $$$(R_i, C_i)$$$. The rest of the cells are white.

You and your friend will alternately take turn playing on this grid, and you are playing in the first turn. In one turn, a player will choose a black cell $$$(r, c)$$$, then toggle cells $$$(r - x, c - y)$$$ for all $$$0 \leq x, y < \min(r, c)$$$. If a cell is toggled, then the cell becomes black if it was a white cell, and the cell becomes white if it was a black cell.

For example, the following illustration shows how the grid changes after a player chooses a black cell $$$(5, 4)$$$ in their turn.

A player who is unable to play on their turn, i.e. no remaining black cells, loses the game, and the opposing player wins the game. If you and your friend are playing optimally, determine who will win the game.

## Input Format
The first line consists of an integer $$$N$$$ ($$$1 \le N \le 200\,000$$$).

Each of the next $$$N$$$ lines consists of two integers $$$R_i$$$ $$$C_i$$$ ($$$1 \leq R_i, C_i \leq 10^9)$$$. For $$$1 \leq i < j \leq N$$$, $$$(R_i, C_i) \neq (R_j, C_j)$$$.

## Output Format
Output FIRST if you will win the game, or SECOND otherwise.

## Examples
```input
2
2 3
2 4
```
```output
FIRST
```
-----
```input
1
2 2
```
```output
SECOND
```
-----
```input
13
1 1
1 4
1 5
2 1
2 4
2 5
4 1
4 2
4 4
5 1
5 2
5 4
5 5
```
```output
SECOND
```

## Note
Explanation for the sample input/output #1

You can start your move by choosing $$$(2, 4)$$$, whose effect was demonstrated in the following illustration.

The remaining black cells are $$$(1, 3)$$$ and $$$(1, 4)$$$, each of which will only toggle itself when chosen. Whichever your friend chooses on the next move, the you can choose the remaining black cell.

Explanation for the sample input/output #2

You have only one cell to choose, and will toggle cells $$$(1, 1)$$$, $$$(1, 2)$$$, $$$(2, 1)$$$, and $$$(2, 2)$$$. Your friend and you will alternately choose the remaining black cells with your friend choosing the last black cell.

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: 1024.0 MB

# Problem
You meet two new friends who are twins. The name of the elder twin is $$$A$$$, which consists of $$$N$$$ characters. While the name of the younger twin is $$$B$$$, which consists of $$$M$$$ characters. It is known that $$$N \leq M$$$.

You want to call each of them with a nickname. For the elder twin, you want to pick any permutation of $$$A$$$ as the nickname. For the younger twin, you want to remove exactly $$$M - N$$$ characters from any permutation of $$$B$$$. Denote the nicknames of the elder twin and the younger twin as $$$A'$$$ and $$$B'$$$, respectively.

You want the nicknames to satisfy the following requirement. For each $$$i$$$ that satisfies $$$1 \leq i \leq N$$$, $$$B'_i$$$ must be equal to either $$$A'_i$$$ or the next letter that follows alphabetically after $$$A'_i$$$ (if such a next letter exists).

Determine the number of different pairs of nicknames $$$(A', B')$$$ that satisfy the requirement. Two pairs of nicknames are considered different if at least one of the nicknames are different. As the result might be large, find the answer modulo $$$998\,244\,353$$$.

## Input Format
The first line consists of two integers $$$N$$$ $$$M$$$ ($$$1 \leq N \leq M \leq 200\,000$$$).

The second line consists of a string $$$A$$$ of length $$$N$$$.

The third line consists of a string $$$B$$$ of length $$$M$$$.

All strings consist of only upper-case letters.

## Output Format
Output a single integer representing number of different pairs $$$(A', B')$$$ that satisfy the requirement, modulo $$$998\,244\,353$$$.

## Examples
```input
3 4
AMA
ANAB
```
```output
9
```
-----
```input
5 8
BINUS
BINANUSA
```
```output
120
```
-----
```input
15 30
BINUSUNIVERSITY
BINANUSANTARAUNIVERSITYJAKARTA
```
```output
151362308
```
-----
```input
4 4
UDIN
ASEP
```
```output
0
```

## Note
Explanation for the sample input/output #1

The $$$9$$$ pairs are:

- (AAM, AAN),
- (AAM, ABN),
- (AAM, BAN),
- (AMA, ANA),
- (AMA, ANB),
- (AMA, BNA),
- (MAA, NAA),
- (MAA, NAB), and
- (MAA, NBA).

Explanation for the sample input/output #2

The $$$120$$$ pairs are the pairs where $$$A'$$$ is a permutation of BINUS and $$$B' = A'$$$.

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: 1024.0 MB

# Problem
You are currently researching a graph traversal algorithm called the Breadth First Search (BFS). Suppose you have an input graph of $$$N$$$ nodes (numbered from $$$1$$$ to $$$N$$$). The graph is represented by an adjacency matrix $$$M$$$, for which node $$$u$$$ can traverse to node $$$v$$$ if $$$M_{u, v}$$$ is $$$1$$$, otherwise it is $$$0$$$. Your algorithm will output the order the nodes are visited in the BFS. The pseudocode of the algorithm is presented as follows.

During your research, you are interested in the following problem. Given an array $$$A$$$ such that $$$A$$$ is a permutation of $$$1$$$ to $$$N$$$ and $$$A_1 = 1$$$. How many simple undirected graph with $$$N$$$ nodes and adjacency matrix $$$M$$$ such that $$$\text{BFS}(M) = A$$$? Since the answer can be very large, calculate the answer modulo $$$998\,244\,353$$$.

A simple graph has no self-loop ($$$M_{i, i} = 0$$$ for $$$1 \leq i \leq N$$$) and there is at most one edge that connects a pair of nodes. In an undirected graph, if node $$$u$$$ is adjacent to node $$$v$$$, then node $$$v$$$ is also adjacent to node $$$u$$$; formally, $$$M_{u, v} = M_{v, u}$$$ for $$$1 \leq u < v \leq N$$$.

Two graphs are considered different if there is an edge that exists in one graph but not the other. In other words, two graphs are considered different if their adjacency matrices are different.

## Input Format
The first line consists of an integer $$$N$$$ ($$$2 \leq N \leq 5000$$$).

The second line consists of $$$N$$$ integers $$$A_i$$$. The array $$$A$$$ is a permutation of $$$1$$$ to $$$N$$$ and $$$A_1 = 1$$$.

## Output Format
Output an integer representing the number of simple undirected graphs with $$$N$$$ nodes and adjacency matrix $$$M$$$ such that $$$\text{BFS}(M) = A$$$. Since the answer can be very large, output the answer modulo $$$998\,244\,353$$$.

## Examples
```input
3
1 2 3
```
```output
3
```
-----
```input
3
1 3 2
```
```output
1
```
-----
```input
5
1 3 2 4 5
```
```output
17
```
-----
```input
11
1 2 3 4 5 6 7 8 9 10 11
```
```output
379394847
```

## Note
Explanation for the sample input/output #1

The following illustration shows all graphs that satisfy the requirements.

Explanation for the sample input/output #2

The only graph that satisfies the requirements is a graph with two edges: one that connects nodes $$$1$$$ and $$$3$$$, and another one that connects nodes $$$3$$$ and $$$2$$$.

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: 1.0 seconds
Memory limit: 1024.0 MB

# Problem
Construct a parentheses sequence consisting of $$$N$$$ characters such that it is balanced and the length of its longest palindromic subsequence (LPS) is exactly $$$K$$$. Determine whether such a construction is possible. If there are several possible sequences, construct any of them.

A parentheses sequence consists of only character ( and ). A parentheses sequence is balanced if each character ( has a corresponding character ) and the pairs of parentheses are properly nested. For example, (), (()), (())(), and ((())()) are balanced. However, )(, ((), and ()) are not balanced.

A sequence is palindromic if it reads the same backwards as forwards. For example, ((, ), ())(, and (()(( are palindromic. However, (), )(, and (()) are not palindromic.

A subsequence can be derived from another sequence by removing zero or more characters without changing the order of the remaining characters. For example, (, ))), ())(, and (())() are subsequence of (())(). However, )(( and ((())) are not subsequence of (())().

The longest palindromic subsequence (LPS) of a sequence is a subsequence with the maximum number of characters, derived from that sequence and it is palindromic. For example, the LPS of sequence (())() is ())(, which can be obtained by removing the second and sixth characters. Therefore, the length of the LPS of (())() is $$$4$$$.

## Input Format
Input consists of two integers $$$N$$$ $$$K$$$ ($$$2 \leq N \leq 2000; 1 \leq K \leq N$$$). $$$N$$$ is an even number.

## Output Format
If there is no such parentheses sequence such that it is balanced and the length of its LPS is exactly $$$K$$$, then output -1.

Otherwise, output a string of $$$N$$$ characters, representing the parentheses sequence. If there are several possible answers, output any of them.

## Examples
```input
6 4
```
```output
(())()
```
-----
```input
6 3
```
```output
(()())
```
-----
```input
4 1
```
```output
-1
```
-----
```input
14 11
```
```output
()((())()())()
```

## Note
Explanation for the sample input/output #2

The LPS of (()()) is either ((( by removing all ) characters, or ))) by removing all ( characters.

The output ((())) also satisfies the requirements.

Explanation for the sample input/output #3

The only possible balanced parentheses sequences are (()) and ()(). The length of the LPS of (()) and ()() are $$$2$$$ and $$$3$$$, respectively.

Explanation for the sample input/output #4

The LPS of ()((())()())() is )())()())(), which can be obtained by removing the first, fourth, and fifth characters.

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: 1024.0 MB

# Problem
You are given a regular $$$N$$$-sided polygon. Label one arbitrary side as side $$$1$$$, then label the next sides in clockwise order as side $$$2$$$, $$$3$$$, $$$\dots$$$, $$$N$$$. There are $$$A_i$$$ special points on side $$$i$$$. These points are positioned such that side $$$i$$$ is divided into $$$A_i + 1$$$ segments with equal length.

For instance, suppose that you have a regular $$$4$$$-sided polygon, i.e., a square. The following illustration shows how the special points are located within each side when $$$A = [3, 1, 4, 6]$$$. The uppermost side is labelled as side $$$1$$$.

You want to create as many non-degenerate triangles as possible while satisfying the following requirements. Each triangle consists of $$$3$$$ distinct special points (not necessarily from different sides) as its corners. Each special point can only become the corner of at most $$$1$$$ triangle. All triangles must not intersect with each other.

Determine the maximum number of non-degenerate triangles that you can create.

A triangle is non-degenerate if it has a positive area.

## Input Format
The first line consists of an integer $$$N$$$ ($$$3 \leq N \leq 200\,000$$$).

The following line consists of $$$N$$$ integers $$$A_i$$$ ($$$1 \leq A_i \leq 2 \cdot 10^9$$$).

## Output Format
Output a single integer representing the maximum number of non-degenerate triangles that you can create.

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

## Note
Explanation for the sample input/output #1

One possible construction which achieves maximum number of non-degenerate triangles can be seen in the following illustration.

Explanation for the sample input/output #2

One possible construction which achieves maximum number of non-degenerate triangles can be seen in the following illustration.

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
As you very well know, this year's funkiest numbers are so called triangular numbers (that is, integers that are representable as $$\frac{k(k+1)}{2}$$, where k is some positive integer), and the coolest numbers are those that are representable as a sum of two triangular numbers.

A well-known hipster Andrew adores everything funky and cool but unfortunately, he isn't good at maths. Given number n, help him define whether this number can be represented by a sum of two triangular numbers (not necessarily different)!

## Input Format
The first input line contains an integer n (1 ≤ n ≤ 109).

## Output Format
Print "YES" (without the quotes), if n can be represented as a sum of two triangular numbers, otherwise print "NO" (without the quotes).

## Examples
```input
256
```
```output
YES
```
-----
```input
512
```
```output
NO
```

## Note
In the first sample number $$256 = \frac{2\cdot3}{2} + \frac{22\cdot23}{2}$$.

In the second sample number 512 can not be represented as a sum of two triangular numbers.

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
In Berland the opposition is going to arrange mass walking on the boulevard. The boulevard consists of n tiles that are lain in a row and are numbered from 1 to n from right to left. The opposition should start walking on the tile number 1 and the finish on the tile number n. During the walk it is allowed to move from right to left between adjacent tiles in a row, and jump over a tile. More formally, if you are standing on the tile number i (i < n - 1), you can reach the tiles number i + 1 or the tile number i + 2 from it (if you stand on the tile number n - 1, you can only reach tile number n). We can assume that all the opposition movements occur instantaneously.

In order to thwart an opposition rally, the Berland bloody regime organized the rain. The tiles on the boulevard are of poor quality and they are rapidly destroyed in the rain. We know that the i-th tile is destroyed after ai days of rain (on day ai tile isn't destroyed yet, and on day ai + 1 it is already destroyed). Of course, no one is allowed to walk on the destroyed tiles! So the walk of the opposition is considered thwarted, if either the tile number 1 is broken, or the tile number n is broken, or it is impossible to reach the tile number n from the tile number 1 if we can walk on undestroyed tiles.

The opposition wants to gather more supporters for their walk. Therefore, the more time they have to pack, the better. Help the opposition to calculate how much time they still have and tell us for how many days the walk from the tile number 1 to the tile number n will be possible.

## Input Format
The first line contains integer n (1 ≤ n ≤ 103) — the boulevard's length in tiles.

The second line contains n space-separated integers ai — the number of days after which the i-th tile gets destroyed (1 ≤ ai ≤ 103).

## Output Format
Print a single number — the sought number of days.

## Examples
```input
4
10 3 5 10
```
```output
5
```
-----
```input
5
10 2 8 3 5
```
```output
5
```

## Note
In the first sample the second tile gets destroyed after day three, and the only path left is 1 → 3 → 4. After day five there is a two-tile gap between the first and the last tile, you can't jump over it.

In the second sample path 1 → 3 → 5 is available up to day five, inclusive. On day six the last tile is destroyed and the walk is thwarted.

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 $$$a_1, a_2, \ldots, a_n$$$. Initially, $$$a_i=i$$$ for each $$$1 \le i \le n$$$.

The operation $$$\texttt{swap}(k)$$$ for an integer $$$k \ge 2$$$ is defined as follows:

- Let $$$d$$$ be the largest divisor$$$^\dagger$$$ of $$$k$$$ which is not equal to $$$k$$$ itself. Then swap the elements $$$a_d$$$ and $$$a_k$$$.

Suppose you perform $$$\texttt{swap}(i)$$$ for each $$$i=2,3,\ldots, n$$$ in this exact order. Find the position of $$$1$$$ in the resulting array. In other words, find such $$$j$$$ that $$$a_j = 1$$$ after performing these operations.

$$$^\dagger$$$ An integer $$$x$$$ is a divisor of $$$y$$$ if there exists an integer $$$z$$$ such that $$$y = x \cdot z$$$.

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

The only line of each test case contains one integer $$$n$$$ ($$$1 \le n \le 10^9$$$) — the length of the array $$$a$$$.

## Output Format
For each test case, output the position of $$$1$$$ in the resulting array.

## Examples
```input
4
1
4
5
120240229
```
```output
1
4
4
67108864
```

## Note
In the first test case, the array is $$$[1]$$$ and there are no operations performed.

In the second test case, $$$a$$$ changes as follows:

- Initially, $$$a$$$ is $$$[1,2,3,4]$$$.
- After performing $$$\texttt{swap}(2)$$$, $$$a$$$ changes to $$$[\underline{2},\underline{1},3,4]$$$ (the elements being swapped are underlined).
- After performing $$$\texttt{swap}(3)$$$, $$$a$$$ changes to $$$[\underline{3},1,\underline{2},4]$$$.
- After performing $$$\texttt{swap}(4)$$$, $$$a$$$ changes to $$$[3,\underline{4},2,\underline{1}]$$$.

Finally, the element $$$1$$$ lies on index $$$4$$$ (that is, $$$a_4 = 1$$$). Thus, the answer is $$$4$$$.

Now solve the problem and return the code.