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: 256.0 MB

# Problem
Tired of boring office work, Denis decided to open a fast food restaurant.

On the first day he made $$$a$$$ portions of dumplings, $$$b$$$ portions of cranberry juice and $$$c$$$ pancakes with condensed milk.

The peculiarity of Denis's restaurant is the procedure of ordering food. For each visitor Denis himself chooses a set of dishes that this visitor will receive. When doing so, Denis is guided by the following rules:

- every visitor should receive at least one dish (dumplings, cranberry juice, pancakes with condensed milk are all considered to be dishes);
- each visitor should receive no more than one portion of dumplings, no more than one portion of cranberry juice and no more than one pancake with condensed milk;
- all visitors should receive different sets of dishes.

What is the maximum number of visitors Denis can feed?

## Input Format
The first line contains an integer $$$t$$$ ($$$1 \le t \le 500$$$) — the number of test cases to solve.

Each of the remaining $$$t$$$ lines contains integers $$$a$$$, $$$b$$$ and $$$c$$$ ($$$0 \leq a, b, c \leq 10$$$) — the number of portions of dumplings, the number of portions of cranberry juice and the number of condensed milk pancakes Denis made.

## Output Format
For each test case print a single integer — the maximum number of visitors Denis can feed.

## Examples
```input
7
1 2 1
0 0 0
9 1 7
2 2 3
2 3 2
3 2 2
4 4 4
```
```output
3
0
4
5
5
5
7
```

## Note
In the first test case of the example, Denis can feed the first visitor with dumplings, give the second a portion of cranberry juice, and give the third visitor a portion of cranberry juice and a pancake with a condensed milk.

In the second test case of the example, the restaurant Denis is not very promising: he can serve no customers.

In the third test case of the example, Denise can serve four visitors. The first guest will receive a full lunch of dumplings, a portion of cranberry juice and a pancake with condensed milk. The second visitor will get only dumplings. The third guest will receive a pancake with condensed milk, and the fourth guest will receive a pancake and a portion of dumplings. Please note that Denis hasn't used all of the prepared products, but is unable to serve more visitors.

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
Nikolay has only recently started in competitive programming, but already qualified to the finals of one prestigious olympiad. There going to be $$$n$$$ participants, one of whom is Nikolay. Like any good olympiad, it consists of two rounds. Tired of the traditional rules, in which the participant who solved the largest number of problems wins, the organizers came up with different rules.

Suppose in the first round participant A took $$$x$$$-th place and in the second round — $$$y$$$-th place. Then the total score of the participant A is sum $$$x + y$$$. The overall place of the participant A is the number of participants (including A) having their total score less than or equal to the total score of A. Note, that some participants may end up having a common overall place. It is also important to note, that in both the first and the second round there were no two participants tying at a common place. In other words, for every $$$i$$$ from $$$1$$$ to $$$n$$$ exactly one participant took $$$i$$$-th place in first round and exactly one participant took $$$i$$$-th place in second round.

Right after the end of the Olympiad, Nikolay was informed that he got $$$x$$$-th place in first round and $$$y$$$-th place in the second round. Nikolay doesn't know the results of other participants, yet he wonders what is the minimum and maximum place he can take, if we consider the most favorable and unfavorable outcome for him. Please help Nikolay to find the answer to this question.

## Input Format
The first line contains an integer $$$t$$$ ($$$1 \le t \le 100$$$) — the number of test cases to solve.

Each of the following $$$t$$$ lines contains integers $$$n$$$, $$$x$$$, $$$y$$$ ($$$1 \leq n \leq 10^9$$$, $$$1 \le x, y \le n$$$) — the number of participants in the olympiad, the place that Nikolay took in the first round and the place that Nikolay took in the second round.

## Output Format
Print two integers — the minimum and maximum possible overall place Nikolay could take.

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

## Note
Explanation for the first example:

Suppose there were 5 participants A-E. Let's denote Nikolay as A. The the most favorable results for Nikolay could look as follows:

However, the results of the Olympiad could also look like this:

In the first case Nikolay would have taken first place, and in the second — third place.

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
Being Santa Claus is very difficult. Sometimes you have to deal with difficult situations.

Today Santa Claus came to the holiday and there were $$$m$$$ children lined up in front of him. Let's number them from $$$1$$$ to $$$m$$$. Grandfather Frost knows $$$n$$$ spells. The $$$i$$$-th spell gives a candy to every child whose place is in the $$$[L_i, R_i]$$$ range. Each spell can be used at most once. It is also known that if all spells are used, each child will receive at most $$$k$$$ candies.

It is not good for children to eat a lot of sweets, so each child can eat no more than one candy, while the remaining candies will be equally divided between his (or her) Mom and Dad. So it turns out that if a child would be given an even amount of candies (possibly zero), then he (or she) will be unable to eat any candies and will go sad. However, the rest of the children (who received an odd number of candies) will be happy.

Help Santa Claus to know the maximum number of children he can make happy by casting some of his spells.

## Input Format
The first line contains three integers of $$$n$$$, $$$m$$$, and $$$k$$$ ($$$1 \leq n \leq 100\,000, 1 \leq m \leq 10^9, 1 \leq k \leq 8$$$) — the number of spells, the number of children and the upper limit on the number of candy a child can get if all spells are used, respectively.

This is followed by $$$n$$$ lines, each containing integers $$$L_i$$$ and $$$R_i$$$ ($$$1 \leq L_i \leq R_i \leq m$$$) — the parameters of the $$$i$$$ spell.

## Output Format
Print a single integer — the maximum number of children that Santa can make happy.

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

## Note
In the first example, Santa should apply the first and third spell. In this case all children will be happy except the third.

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
Vasya had three strings $$$a$$$, $$$b$$$ and $$$s$$$, which consist of lowercase English letters. The lengths of strings $$$a$$$ and $$$b$$$ are equal to $$$n$$$, the length of the string $$$s$$$ is equal to $$$m$$$.

Vasya decided to choose a substring of the string $$$a$$$, then choose a substring of the string $$$b$$$ and concatenate them. Formally, he chooses a segment $$$[l_1, r_1]$$$ ($$$1 \leq l_1 \leq r_1 \leq n$$$) and a segment $$$[l_2, r_2]$$$ ($$$1 \leq l_2 \leq r_2 \leq n$$$), and after concatenation he obtains a string $$$a[l_1, r_1] + b[l_2, r_2] = a_{l_1} a_{l_1 + 1} \ldots a_{r_1} b_{l_2} b_{l_2 + 1} \ldots b_{r_2}$$$.

Now, Vasya is interested in counting number of ways to choose those segments adhering to the following conditions:

- segments $$$[l_1, r_1]$$$ and $$$[l_2, r_2]$$$ have non-empty intersection, i.e. there exists at least one integer $$$x$$$, such that $$$l_1 \leq x \leq r_1$$$ and $$$l_2 \leq x \leq r_2$$$;
- the string $$$a[l_1, r_1] + b[l_2, r_2]$$$ is equal to the string $$$s$$$.

## Input Format
The first line contains integers $$$n$$$ and $$$m$$$ ($$$1 \leq n \leq 500\,000, 2 \leq m \leq 2 \cdot n$$$) — the length of strings $$$a$$$ and $$$b$$$ and the length of the string $$$s$$$.

The next three lines contain strings $$$a$$$, $$$b$$$ and $$$s$$$, respectively. The length of the strings $$$a$$$ and $$$b$$$ is $$$n$$$, while the length of the string $$$s$$$ is $$$m$$$.

All strings consist of lowercase English letters.

## Output Format
Print one integer — the number of ways to choose a pair of segments, which satisfy Vasya's conditions.

## Examples
```input
6 5
aabbaa
baaaab
aaaaa
```
```output
4
```
-----
```input
5 4
azaza
zazaz
azaz
```
```output
11
```
-----
```input
9 12
abcabcabc
xyzxyzxyz
abcabcayzxyz
```
```output
2
```

## Note
Let's list all the pairs of segments that Vasya could choose in the first example:

1. $$$[2, 2]$$$ and $$$[2, 5]$$$;
2. $$$[1, 2]$$$ and $$$[2, 4]$$$;
3. $$$[5, 5]$$$ and $$$[2, 5]$$$;
4. $$$[5, 6]$$$ and $$$[3, 5]$$$;

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: 1.0 seconds
Memory limit: 256.0 MB

# Problem
$$$n$$$ students are taking an exam. The highest possible score at this exam is $$$m$$$. Let $$$a_{i}$$$ be the score of the $$$i$$$-th student. You have access to the school database which stores the results of all students.

You can change each student's score as long as the following conditions are satisfied:

- All scores are integers
- $$$0 \leq a_{i} \leq m$$$
- The average score of the class doesn't change.

You are student $$$1$$$ and you would like to maximize your own score.

Find the highest possible score you can assign to yourself such that all conditions are satisfied.

## Input Format
Each test contains multiple test cases.

The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 200$$$). The description of the test cases follows.

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \leq n \leq 10^{3}$$$, $$$1 \leq m \leq 10^{5}$$$)  — the number of students and the highest possible score respectively.

The second line of each testcase contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$ 0 \leq a_{i} \leq m$$$)  — scores of the students.

## Output Format
For each testcase, output one integer  — the highest possible score you can assign to yourself such that both conditions are satisfied._

## Examples
```input
2
4 10
1 2 3 4
4 5
1 2 3 4
```
```output
10
5
```

## Note
In the first case, $$$a = [1,2,3,4] $$$, with average of $$$2.5$$$. You can change array $$$a$$$ to $$$[10,0,0,0]$$$. Average remains $$$2.5$$$, and all conditions are satisfied.

In the second case, $$$0 \leq a_{i} \leq 5$$$. You can change $$$a$$$ to $$$[5,1,1,3]$$$. You cannot increase $$$a_{1}$$$ further as it will violate condition $$$0\le a_i\le m$$$.

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 special jigsaw puzzle consisting of $$$n\cdot m$$$ identical pieces. Every piece has three tabs and one blank, as pictured below.

The jigsaw puzzle is considered solved if the following conditions hold:

1. The pieces are arranged into a grid with $$$n$$$ rows and $$$m$$$ columns.
2. For any two pieces that share an edge in the grid, a tab of one piece fits perfectly into a blank of the other piece.

Through rotation and translation of the pieces, determine if it is possible to solve the jigsaw puzzle.

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

Each test case contains two integers $$$n$$$ and $$$m$$$ ($$$1 \le n,m \le 10^5$$$).

## Output Format
For each test case output a single line containing "YES" if it is possible to solve the jigsaw puzzle, or "NO" otherwise. You can print each letter in any case (upper or lower).

## Examples
```input
3
1 3
100000 100000
2 2
```
```output
YES
NO
YES
```

## Note
For the first test case, this is an example solution:

For the second test case, we can show that no solution exists.

For the third test case, this is an example solution:

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
A card pyramid of height $$$1$$$ is constructed by resting two cards against each other. For $$$h>1$$$, a card pyramid of height $$$h$$$ is constructed by placing a card pyramid of height $$$h-1$$$ onto a base. A base consists of $$$h$$$ pyramids of height $$$1$$$, and $$$h-1$$$ cards on top. For example, card pyramids of heights $$$1$$$, $$$2$$$, and $$$3$$$ look as follows:

You start with $$$n$$$ cards and build the tallest pyramid that you can. If there are some cards remaining, you build the tallest pyramid possible with the remaining cards. You repeat this process until it is impossible to build another pyramid. In the end, how many pyramids will you have constructed?

## Input Format
Each test consists of multiple test cases. The first line contains a single integer $$$t$$$ ($$$1\le t\le 1000$$$) — the number of test cases. Next $$$t$$$ lines contain descriptions of test cases.

Each test case contains a single integer $$$n$$$ ($$$1\le n\le 10^9$$$) — the number of cards.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$10^9$$$.

## Output Format
For each test case output a single integer — the number of pyramids you will have constructed in the end.

## Examples
```input
5
3
14
15
24
1
```
```output
1
2
1
3
0
```

## Note
In the first test, you construct a pyramid of height $$$1$$$ with $$$2$$$ cards. There is $$$1$$$ card remaining, which is not enough to build a pyramid.

In the second test, you build two pyramids, each of height $$$2$$$, with no cards remaining.

In the third test, you build one pyramid of height $$$3$$$, with no cards remaining.

In the fourth test, you build one pyramid of height $$$3$$$ with $$$9$$$ cards remaining. Then you build a pyramid of height $$$2$$$ with $$$2$$$ cards remaining. Then you build a final pyramid of height $$$1$$$ with no cards remaining.

In the fifth test, one card is not enough to build any pyramids.

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
Due to the coronavirus pandemic, city authorities obligated citizens to keep a social distance. The mayor of the city Semyon wants to light up Gluharniki park so that people could see each other even at night to keep the social distance.

The park is a rectangular table with $$$n$$$ rows and $$$m$$$ columns, where the cells of the table are squares, and the boundaries between the cells are streets. External borders are also streets. Every street has length $$$1$$$. For example, park with $$$n=m=2$$$ has $$$12$$$ streets.

You were assigned to develop a plan for lighting the park. You can put lanterns in the middle of the streets. The lamp lights two squares near it (or only one square if it stands on the border of the park).

The park sizes are: $$$n=4$$$, $$$m=5$$$. The lighted squares are marked yellow. Please note that all streets have length $$$1$$$. Lanterns are placed in the middle of the streets. In the picture not all the squares are lit.

Semyon wants to spend the least possible amount of money on lighting but also wants people throughout the park to keep a social distance. So he asks you to find the minimum number of lanterns that are required to light all the squares.

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

Each test case is a line containing two integers $$$n$$$, $$$m$$$ ($$$1 \le n, m \le 10^4$$$) — park sizes.

## Output Format
Print $$$t$$$ answers to the test cases. Each answer must be a single integer — the minimum number of lanterns that are required to light all the squares.

## Examples
```input
5
1 1
1 3
2 2
3 3
5 3
```
```output
1
2
2
5
8
```

## Note
Possible optimal arrangement of the lanterns for the $$$2$$$-nd test case of input data example:

Possible optimal arrangement of the lanterns for the $$$3$$$-rd test case of input data example:

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
During the quarantine, Sicromoft has more free time to create the new functions in "Celex-2021". The developers made a new function GAZ-GIZ, which infinitely fills an infinite table to the right and down from the upper left corner as follows:

The developers of the SUM function don't sleep either. Because of the boredom, they teamed up with the developers of the RAND function, so they added the ability to calculate the sum on an arbitrary path from one cell to another, moving down or right. Formally, from the cell $$$(x,y)$$$ in one step you can move to the cell $$$(x+1, y)$$$ or $$$(x, y+1)$$$.

After another Dinwows update, Levian started to study "Celex-2021" (because he wants to be an accountant!). After filling in the table with the GAZ-GIZ function, he asked you to calculate the quantity of possible different amounts on the path from a given cell $$$(x_1, y_1)$$$ to another given cell $$$(x_2, y_2$$$), if you can only move one cell down or right.

Formally, consider all the paths from the cell $$$(x_1, y_1)$$$ to cell $$$(x_2, y_2)$$$ such that each next cell in the path is located either to the down or to the right of the previous one. Calculate the number of different sums of elements for all such paths.

## Input Format
The first line contains one integer $$$t$$$ ($$$1 \le t \le 57179$$$) — the number of test cases.

Each of the following $$$t$$$ lines contains four natural numbers $$$x_1$$$, $$$y_1$$$, $$$x_2$$$, $$$y_2$$$ ($$$1 \le x_1 \le x_2 \le 10^9$$$, $$$1 \le y_1 \le y_2 \le 10^9$$$) — coordinates of the start and the end cells.

## Output Format
For each test case, in a separate line, print the number of possible different sums on the way from the start cell to the end cell.

## Examples
```input
4
1 1 2 2
1 2 2 4
179 1 179 100000
5 7 5 7
```
```output
2
3
1
1
```

## Note
In the first test case there are two possible sums: $$$1+2+5=8$$$ and $$$1+3+5=9$$$.

Now solve the problem and return the code.