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
Some dwarves that are finishing the StUDY (State University for Dwarven Youngsters) Bachelor courses, have been told "no genome, no degree". That means that all dwarves should write a thesis on genome. Dwarven genome is far from simple. It is represented by a string that consists of lowercase Latin letters.

Dwarf Misha has already chosen the subject for his thesis: determining by two dwarven genomes, whether they belong to the same race. Two dwarves belong to the same race if we can swap two characters in the first dwarf's genome and get the second dwarf's genome as a result. Help Dwarf Misha and find out whether two gnomes belong to the same race or not.

## Input Format
The first line contains the first dwarf's genome: a non-empty string, consisting of lowercase Latin letters.

The second line contains the second dwarf's genome: a non-empty string, consisting of lowercase Latin letters.

The number of letters in each genome doesn't exceed 105. It is guaranteed that the strings that correspond to the genomes are different. The given genomes may have different length.

## Output Format
Print "YES", if the dwarves belong to the same race. Otherwise, print "NO".

## Examples
```input
ab
ba
```
```output
YES
```
-----
```input
aa
ab
```
```output
NO
```

## Note
- First example: you can simply swap two letters in string "ab". So we get "ba".
- Second example: we can't change string "aa" into string "ab", because "aa" does not contain letter "b".

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
A bracket sequence is a string consisting of characters '(' and/or ')'. A regular bracket sequence is a bracket sequence that can be transformed into a correct arithmetic expression by inserting characters '1' and '+' between the original characters of the sequence. For example:

- bracket sequences "()()" and "(())" are regular (they can be transformed into "(1)+(1)" and "((1+1)+1)", respectively);
- bracket sequences ")(", "(" and ")" are not regular.

You are given a bracket sequence $$$s$$$; let's define its length as $$$n$$$. Your task is to find a regular bracket sequence $$$t$$$ of length $$$2n$$$ such that $$$s$$$ does not occur in $$$t$$$ as a contiguous substring, or report that there is no such sequence.

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

The only line of each test case contains a string $$$s$$$ ($$$2 \le |s| \le 50$$$), consisting of characters "(" and/or ")".

## Output Format
For each test case, print the answer to it. If there is no required regular bracket sequence, print NO in a separate line. Otherwise, print YES in the first line, and the required regular bracket sequence $$$t$$$ itself in the second line. If there are multiple answers — you may print any of them.

## Examples
```input
4
)(
(()
()
))()
```
```output
YES
(())
YES
()()()
NO
YES
()(()())
```

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
Monocarp is going to make a purchase with cost of exactly $$$m$$$ burles.

He has two types of coins, in the following quantities:

- coins worth $$$1$$$ burle: $$$a_1$$$ regular coins and infinitely many fancy coins;
- coins worth $$$k$$$ burles: $$$a_k$$$ regular coins and infinitely many fancy coins.

Monocarp wants to make his purchase in such a way that there's no change — the total worth of provided coins is exactly $$$m$$$. He can use both regular and fancy coins. However, he wants to spend as little fancy coins as possible.

What's the smallest total number of fancy coins he can use to make a purchase?

## Input Format
The first line contains a single integer $$$t$$$ ($$$1 \le t \le 3 \cdot 10^4$$$) — the number of testcases.

The only line of each testcase contains four integers $$$m, k, a_1$$$ and $$$a_k$$$ ($$$1 \le m \le 10^8$$$; $$$2 \le k \le 10^8$$$; $$$0 \le a_1, a_k \le 10^8$$$) — the cost of the purchase, the worth of the second type of coin and the amounts of regular coins of both types, respectively.

## Output Format
For each testcase, print a single integer — the smallest total number of fancy coins Monocarp can use to make a purchase.

## Examples
```input
4
11 3 0 0
11 3 20 20
11 3 6 1
100000000 2 0 0
```
```output
5
0
1
50000000
```

## Note
In the first testcase, there are no regular coins of either type. Monocarp can use $$$2$$$ fancy coins worth $$$1$$$ burle and $$$3$$$ fancy coins worth $$$3$$$ (since $$$k=3$$$) burles to get $$$11$$$ total burles with $$$5$$$ total fancy coins.

In the second testcase, Monocarp has a lot of regular coins of both types. He can use $$$11$$$ regular coins worth $$$1$$$ burle, for example. Notice that Monocarp doesn't have to minimize the total number of used coins. That way he uses $$$0$$$ fancy coins.

In the third testcase, Monocarp can use $$$5$$$ regular coins worth $$$1$$$ burle and $$$1$$$ regular coin worth $$$3$$$ burles. That will get him to $$$8$$$ total burles when he needs $$$11$$$. So, $$$1$$$ fancy coin worth $$$3$$$ burles is enough.

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 binary string $$$s$$$ (a binary string is a string consisting of characters 0 and/or 1).

Let's call a binary string balanced if the number of subsequences 01 (the number of indices $$$i$$$ and $$$j$$$ such that $$$1 \le i < j \le n$$$, $$$s_i=0$$$ and $$$s_j=1$$$) equals to the number of subsequences 10 (the number of indices $$$k$$$ and $$$l$$$ such that $$$1 \le k < l \le n$$$, $$$s_k=1$$$ and $$$s_l=0$$$) in it.

For example, the string 1000110 is balanced, because both the number of subsequences 01 and the number of subsequences 10 are equal to $$$6$$$. On the other hand, 11010 is not balanced, because the number of subsequences 01 is $$$1$$$, but the number of subsequences 10 is $$$5$$$.

You can perform the following operation any number of times: choose two characters in $$$s$$$ and swap them. Your task is to calculate the minimum number of operations to make the string $$$s$$$ balanced.

## Input Format
The only line contains the string $$$s$$$ ($$$3 \le |s| \le 100$$$) consisting of characters 0 and/or 1.

Additional constraint on the input: the string $$$s$$$ can be made balanced.

## Output Format
Print a single integer — the minimum number of swap operations to make the string $$$s$$$ balanced.

## Examples
```input
101
```
```output
0
```
-----
```input
1000110
```
```output
0
```
-----
```input
11010
```
```output
1
```
-----
```input
11001100
```
```output
2
```

## Note
In the first example, the string is already balanced, the number of both 01 and 10 is equal to $$$1$$$.

In the second example, the string is already balanced, the number of both 01 and 10 is equal to $$$6$$$.

In the third example, one of the possible answers is the following one: 11010 $$$\rightarrow$$$ 01110. After that, the number of both 01 and 10 is equal to $$$3$$$.

In the fourth example, one of the possible answers is the following one: 11001100 $$$\rightarrow$$$ 11001010 $$$\rightarrow$$$ 11000011. After that, the number of both 01 and 10 is equal to $$$8$$$.

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: 512.0 MB

# Problem
You are given a string $$$s$$$, consisting of lowercase Latin letters.

There's a cursor, initially placed between two adjacent letters of the string. The cursor can't be placed before the first or after the last letter.

In one move, you can perform one of three actions:

- move a cursor one position to the left (if that doesn't place it before the first letter);
- move a cursor one position to the right (if that doesn't place it after the last letter);
- let $$$x$$$ be the letter immediately to the left of the cursor, and $$$y$$$ be the letter immediately to the right of the cursor. Choose any pair of adjacent letters in the string such that the left of them is $$$x$$$ and the right of them is $$$y$$$, and move the cursor to the position between these two letters.

You are asked $$$m$$$ queries. In each query, you are given two integers $$$f$$$ and $$$t$$$, which correspond to two valid positions of the cursor in the string. In response to the query, you have to calculate the minimum number of operations required to move the cursor from position $$$f$$$ to position $$$t$$$.

## Input Format
The first line contains a string $$$s$$$ ($$$2 \le |s| \le 5 \cdot 10^4$$$), consisting of lowercase Latin letters.

The second line contains a single integer $$$m$$$ ($$$1 \le m \le 5 \cdot 10^4$$$) — the number of queries.

The $$$i$$$-th of the next $$$m$$$ lines contains two integers $$$f$$$ and $$$t$$$ ($$$1 \le f, t \le |s| - 1$$$) — the initial and the target positions of the cursor in the $$$i$$$-th query. Position $$$x$$$ means that the cursor is between the $$$x$$$-th and the $$$(x+1)$$$-st letters of the string ($$$1$$$-indexed).

## Output Format
For each query, print the minimum number of actions required to move the cursor from position $$$f$$$ to position $$$t$$$.

## Examples
```input
codecode
3
1 7
3 5
3 6
```
```output
3
2
2
```
-----
```input
abcdefghij
3
1 9
9 1
5 5
```
```output
8
8
0
```
-----
```input
aaaaaaaaaaaa
4
10 8
3 7
6 1
11 11
```
```output
1
1
1
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: 10.0 seconds
Memory limit: 512.0 MB

# Problem
You are given $$$2n$$$ tuples of values $$$(a, b, c)$$$, where $$$a$$$ and $$$b$$$ are positive integers and $$$c$$$ is a bracket '(' or ')'. Exactly $$$n$$$ tuples have $$$c = $$$'(' and the other $$$n$$$ tuples have $$$c =$$$ ')'.

You are asked to choose two positive values $$$x$$$ and $$$y$$$ ($$$x > 0$$$ and $$$y > 0$$$; not necessarily integers) and sort the tuples in the increasing value of $$$a \cdot x + b \cdot y$$$. If several tuples have the same value, you can place them in any order among themselves.

Is it possible to choose such $$$x$$$ and $$$y$$$ that taking brackets $$$c$$$ from the tuples in the resulting order produces a regular bracket sequence?

A regular bracket sequence is a bracket sequence that can be transformed into a correct arithmetic expression by inserting characters "1" and "+" between the original characters of the sequence.

## Input Format
The first line contains a single integer $$$t$$$ ($$$1 \le t \le 1500$$$) — the number of testcases.

The first line of each testcase contains a single integer $$$n$$$ ($$$1 \le n \le 1500$$$).

The $$$i$$$-th of the next $$$2n$$$ lines contains two integers $$$a_i$$$ and $$$b_i$$$ ($$$1 \le a_i, b_i \le 10^6$$$) and a character $$$c_i$$$ ($$$c_i = $$$'(' or $$$c_i = $$$')'). Exactly $$$n$$$ tuples have $$$c_i = $$$'(' and the other $$$n$$$ tuples have $$$c_i =$$$ ')'.

The sum of $$$n$$$ over all testcases doesn't exceed $$$1500$$$.

## Output Format
For each testcase, print "YES" if it's possible to choose such $$$x$$$ and $$$y$$$ that taking brackets $$$c$$$ from the tuples in the resulting order produces a regular bracket sequence. Otherwise, print "NO".

## Examples
```input
4
1
1 4 (
2 3 )
1
1 2 )
3 4 (
4
16 5 (
12 3 (
19 6 )
4 10 )
3 10 )
19 11 (
19 7 )
7 14 (
4
16 8 (
11 9 )
20 10 )
20 19 )
2 13 (
18 7 (
15 19 )
5 6 (
```
```output
YES
NO
NO
YES
```

## Note
In the first testcase, you can choose $$$x = 10, y = 0.1$$$. The values for tuples will be $$$10 \cdot 1 + 0.1 \cdot 4 = 10.4$$$ and $$$10 \cdot 2 + 0.1 \cdot 3 = 20.3$$$. Thus, the first tuple will go first, then the second one, making the bracket sequence "()", which is regular.

In the second testcase, you can't choose positive $$$x$$$ and $$$y$$$ such that the opening brackets gets assigned a value less or equal to the value of the closing bracket.

In the fourth testcase, you can choose $$$x = 0.6, y = 0.55$$$. The bracket sequence 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
You are given three integers $$$x$$$, $$$y$$$, and $$$n$$$.

Your task is to construct an array $$$a$$$ consisting of $$$n$$$ integers which satisfies the following conditions:

1. $$$a_1=x$$$, $$$a_n=y$$$;
2. $$$a$$$ is strictly increasing (i.e. $$$a_1 < a_2 < \ldots < a_n$$$);
3. if we denote $$$b_i=a_{i+1}-a_{i}$$$ for $$$1 \leq i \leq n-1$$$, then $$$b$$$ is strictly decreasing (i.e. $$$b_1 > b_2 > \ldots > b_{n-1}$$$).

If there is no such array $$$a$$$, print a single integer $$$-1$$$.

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

The only line of each test case contains three integers $$$x$$$, $$$y$$$, $$$n$$$ ($$$1 \le x < y \le 1000,3 \le n \le 1000$$$).

## Output Format
For each test case, output $$$n$$$ integers $$$a_1,a_2,\ldots,a_n$$$. If there are multiple solutions, print any of them.

If there is no solution, print a single integer $$$-1$$$.

## Examples
```input
3
1 4 3
1 3 3
100 200 4
```
```output
1 3 4
-1
100 150 180 200
```

## Note
In the first test case, $$$a=[1,3,4]$$$, which is strictly increasing. Next, $$$b_1=a_2-a_1=3-1=2$$$, $$$b_2=a_3-a_2=4-3=1$$$, thus $$$b=[2,1]$$$, which is strictly decreasing.

In the second test case, there is no array $$$a$$$ that satisfies all the conditions above.

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 integer $$$x$$$. Your task is to reduce $$$x$$$ to $$$1$$$.

To do that, you can do the following operation:

- select a divisor $$$d$$$ of $$$x$$$, then change $$$x$$$ to $$$x-d$$$, i.e. reduce $$$x$$$ by $$$d$$$. (We say that $$$d$$$ is a divisor of $$$x$$$ if $$$d$$$ is an positive integer and there exists an integer $$$q$$$ such that $$$x = d \cdot q$$$.)

There is an additional constraint: you cannot select the same value of $$$d$$$ more than twice.

For example, for $$$x=5$$$, the following scheme is invalid because $$$1$$$ is selected more than twice: $$$5\xrightarrow{-1}4\xrightarrow{-1}3\xrightarrow{-1}2\xrightarrow{-1}1$$$. The following scheme is however a valid one: $$$5\xrightarrow{-1}4\xrightarrow{-2}2\xrightarrow{-1}1$$$.

Output any scheme which reduces $$$x$$$ to $$$1$$$ with at most $$$1000$$$ operations. It can be proved that such a scheme always exists.

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

The only line of each test case contains a single integer $$$x$$$ ($$$2\le x \le 10^{9}$$$).

## Output Format
For each test case, output two lines.

The first line should contain an integer $$$k$$$ ($$$1 \le k \le 1001$$$).

The next line should contain $$$k$$$ integers $$$a_1,a_2,\ldots,a_k$$$, which satisfy the following:

- $$$a_1=x$$$;
- $$$a_k=1$$$;
- for each $$$2 \le i \le k$$$, the value $$$(a_{i-1}-a_i)$$$ is a divisor of $$$a_{i-1}$$$. Each number may occur as a divisor at most twice.

## Examples
```input
3
3
5
14
```
```output
3
3 2 1
4
5 4 2 1
6
14 12 6 3 2 1
```

## Note
In the first test case, we use the following scheme: $$$3\xrightarrow{-1}2\xrightarrow{-1}1$$$.

In the second test case, we use the following scheme: $$$5\xrightarrow{-1}4\xrightarrow{-2}2\xrightarrow{-1}1$$$.

In the third test case, we use the following scheme: $$$14\xrightarrow{-2}12\xrightarrow{-6}6\xrightarrow{-3}3\xrightarrow{-1}2\xrightarrow{-1}1$$$.

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
You are given a sequence of integers $$$a$$$ of length $$$2n$$$. You have to split these $$$2n$$$ integers into $$$n$$$ pairs; each pair will represent the coordinates of a point on a plane. Each number from the sequence $$$a$$$ should become the $$$x$$$ or $$$y$$$ coordinate of exactly one point. Note that some points can be equal.

After the points are formed, you have to choose a path $$$s$$$ that starts from one of these points, ends at one of these points, and visits all $$$n$$$ points at least once.

The length of path $$$s$$$ is the sum of distances between all adjacent points on the path. In this problem, the distance between two points $$$(x_1, y_1)$$$ and $$$(x_2, y_2)$$$ is defined as $$$|x_1-x_2| + |y_1-y_2|$$$.

Your task is to form $$$n$$$ points and choose a path $$$s$$$ in such a way that the length of path $$$s$$$ is minimized.

## Input Format
The first line contains a single integer $$$t$$$ ($$$1 \le t \le 100$$$) — the number of testcases.

The first line of each testcase contains a single integer $$$n$$$ ($$$2 \le n \le 100$$$) — the number of points to be formed.

The next line contains $$$2n$$$ integers $$$a_1, a_2, \dots, a_{2n}$$$ ($$$0 \le a_i \le 1\,000$$$) — the description of the sequence $$$a$$$.

## Output Format
For each testcase, print the minimum possible length of path $$$s$$$ in the first line.

In the $$$i$$$-th of the following $$$n$$$ lines, print two integers $$$x_i$$$ and $$$y_i$$$ — the coordinates of the point that needs to be visited at the $$$i$$$-th position.

If there are multiple answers, print any of them.

## Examples
```input
2
2
15 1 10 5
3
10 30 20 20 30 10
```
```output
9
10 1
15 5
20
20 20
10 30
10 30
```

## Note
In the first testcase, for instance, you can form points $$$(10, 1)$$$ and $$$(15, 5)$$$ and start the path $$$s$$$ from the first point and end it at the second point. Then the length of the path will be $$$|10 - 15| + |1 - 5| = 5 + 4 = 9$$$.

In the second testcase, you can form points $$$(20, 20)$$$, $$$(10, 30)$$$, and $$$(10, 30)$$$, and visit them in that exact order. Then the length of the path will be $$$|20 - 10| + |20 - 30| + |10 - 10| + |30 - 30| = 10 + 10 + 0 + 0 = 20$$$.

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
A ticket is a non-empty string of digits from $$$1$$$ to $$$9$$$.

A lucky ticket is such a ticket that:

- it has an even length;
- the sum of digits in the first half is equal to the sum of digits in the second half.

You are given $$$n$$$ ticket pieces $$$s_1, s_2, \dots, s_n$$$. How many pairs $$$(i, j)$$$ (for $$$1 \le i, j \le n$$$) are there such that $$$s_i + s_j$$$ is a lucky ticket? Note that it's possible that $$$i=j$$$.

Here, the + operator denotes the concatenation of the two strings. For example, if $$$s_i$$$ is 13, and $$$s_j$$$ is 37, then $$$s_i + s_j$$$ is 1337.

## Input Format
The first line contains a single integer $$$n$$$ ($$$1 \le n \le 2 \cdot 10^5$$$) — the number of ticket pieces.

The second line contains $$$n$$$ non-empty strings $$$s_1, s_2, \dots, s_n$$$, each of length at most $$$5$$$ and consisting only of digits from $$$1$$$ to $$$9$$$.

## Output Format
Print a single integer — the number of pairs $$$(i, j)$$$ (for $$$1 \le i, j \le n$$$) such that $$$s_i + s_j$$$ is a lucky ticket.

## Examples
```input
10
5 93746 59 3746 593 746 5937 46 59374 6
```
```output
20
```
-----
```input
5
2 22 222 2222 22222
```
```output
13
```
-----
```input
3
1 1 1
```
```output
9
```

Now solve the problem and return the code.