apaz/codeforces-genesys huggingface dataset:
You are an expert competitive programmer. You will be given a problem statement, test case constraints and example test inputs and outputs. Please reason step by step about the solution (that must respect memory and time limits), then provide a complete implementation in python3.

Your solution must read input from standard input (cin), write output to standard output (cout).
Do not include any debug prints or additional output.

Put your final solution within a single code block:
```python
<your code here>
```

Execution time limit: 2.0 seconds
Memory limit: 256.0 MB

# Problem
You are given two strings $$$a$$$ and $$$b$$$, consisting of lowercase Latin letters.

A template $$$t$$$ is string, consisting of lowercase Latin letters and asterisks (character '*'). A template is called asterisk-minor if the number of asterisks in it is less than or equal to the number of letters in it.

A string $$$s$$$ is said to be matching a template $$$t$$$ if you can replace each asterisk in $$$t$$$ with a string of lowercase Latin letters (possibly, an empty string) so that it becomes equal to $$$s$$$.

Find an asterisk-minor template such that both $$$a$$$ and $$$b$$$ match it, or report that such a template doesn't exist. If there are multiple answers, print any of them.

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

The first line of each testcase contains a string $$$a$$$ ($$$1 \le |a| \le 50$$$, where $$$|a|$$$ is the length of $$$a$$$), consisting of lowercase Latin letters.

The second line contains a string $$$b$$$ ($$$1 \le |b| \le 50$$$), consisting of lowercase Latin letters.

## Output Format
For each testcase, output "NO", if there doesn't exist an asterisk-minor template that both $$$a$$$ and $$$b$$$ match. Otherwise, print "YES" in the first line and the template in the second line. If there are multiple answers, print any of them.

A template should consist only of lowercase Latin letters and asterisks (character '*'). The number of asterisks should be less than or equal to the number of letters.

## Examples
```input
6
aaab
zzzb
codeforces
atcoder
codeforces
tokitlx
aaaa
aaaaaa
abcd
abcd
c
f
```
```output
YES
*b
YES
*co*
NO
YES
a*a*a*a
YES
abcd
NO
```

## Note
In the first testcase, for a template "*b", you can replace the only asterisk with "aaa" to get "aaab" (which is equal to $$$a$$$) or with "zzz" to get "zzzb" (which is equal to $$$b$$$).

In the third testcase, a template "*o*" is not asterisk-minor, as it contains more asterisks than letters. There are no asterisk-minor templates that both $$$a$$$ and $$$b$$$ match.

In the fourth testcase, for a template "a*a*a*a", you can replace all asterisks with empty strings to get "aaaa" (which is equal to $$$a$$$) or two of them with "a" and two of them with an empty string to get "aaaaaa" (which is equal to $$$b$$$).

In the fifth testcase, there are no asterisks in a template "abcd", so only "abcd" can match it (which is coincidentally both $$$a$$$ and $$$b$$$).

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: 512.0 MB

# Problem
A set of positive integers $$$S$$$ is called beautiful if, for every two integers $$$x$$$ and $$$y$$$ from this set, either $$$x$$$ divides $$$y$$$ or $$$y$$$ divides $$$x$$$ (or both).

You are given two integers $$$l$$$ and $$$r$$$. Consider all beautiful sets consisting of integers not less than $$$l$$$ and not greater than $$$r$$$. You have to print two numbers:

- the maximum possible size of a beautiful set where all elements are from $$$l$$$ to $$$r$$$;
- the number of beautiful sets consisting of integers from $$$l$$$ to $$$r$$$ with the maximum possible size.

Since the second number can be very large, print it modulo $$$998244353$$$.

## Input Format
The first line contains one integer $$$t$$$ ($$$1 \le t \le 2 \cdot 10^4$$$) — the number of test cases.

Each test case consists of one line containing two integers $$$l$$$ and $$$r$$$ ($$$1 \le l \le r \le 10^6$$$).

## Output Format
For each test case, print two integers — the maximum possible size of a beautiful set consisting of integers from $$$l$$$ to $$$r$$$, and the number of such sets with maximum possible size. Since the second number can be very large, print it modulo $$$998244353$$$.

## Examples
```input
4
3 11
13 37
1 22
4 100
```
```output
2 4
2 6
5 1
5 7
```

## Note
In the first test case, the maximum possible size of a beautiful set with integers from $$$3$$$ to $$$11$$$ is $$$2$$$. There are $$$4$$$ such sets which have the maximum possible size:

- $$$\{ 3, 6 \}$$$;
- $$$\{ 3, 9 \}$$$;
- $$$\{ 4, 8 \}$$$;
- $$$\{ 5, 10 \}$$$.

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: 256.0 MB

# Problem
Let's call a triple of positive integers ($$$a, b, n$$$) strange if the equality $$$\frac{an}{nb} = \frac{a}{b}$$$ holds, where $$$an$$$ is the concatenation of $$$a$$$ and $$$n$$$ and $$$nb$$$ is the concatenation of $$$n$$$ and $$$b$$$. For the purpose of concatenation, the integers are considered without leading zeroes.

For example, if $$$a = 1$$$, $$$b = 5$$$ and $$$n = 9$$$, then the triple is strange, because $$$\frac{19}{95} = \frac{1}{5}$$$. But $$$a = 7$$$, $$$b = 3$$$ and $$$n = 11$$$ is not strange, because $$$\frac{711}{113} \ne \frac{7}{3}$$$.

You are given three integers $$$A$$$, $$$B$$$ and $$$N$$$. Calculate the number of strange triples $$$(a, b, n$$$), such that $$$1 \le a < A$$$, $$$1 \le b < B$$$ and $$$1 \le n < N$$$.

## Input Format
The only line contains three integers $$$A$$$, $$$B$$$ and $$$N$$$ ($$$1 \le A, B \le 10^5$$$; $$$1 \le N \le 10^9$$$).

## Output Format
Print one integer — the number of strange triples $$$(a, b, n$$$) such that $$$1 \le a < A$$$, $$$1 \le b < B$$$ and $$$1 \le n < N$$$.

## Examples
```input
5 6 10
```
```output
7
```
-----
```input
10 10 100
```
```output
29
```
-----
```input
1 10 25
```
```output
0
```
-----
```input
4242 6969 133333337
```
```output
19536
```
-----
```input
94841 47471 581818184
```
```output
98715
```

## Note
In the first example, there are $$$7$$$ strange triples: $$$(1, 1, 1$$$), ($$$1, 4, 6$$$), ($$$1, 5, 9$$$), ($$$2, 2, 2$$$), ($$$2, 5, 6$$$), ($$$3, 3, 3$$$) and ($$$4, 4, 4$$$).

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
The Zoo in the Grid Kingdom is represented by an infinite grid. The Zoo has n observation binoculars located at the OX axis. For each i between 1 and n, inclusive, there exists a single binocular located at the point with coordinates (i, 0). There are m flamingos in the Zoo, located at points with positive coordinates. The flamingos are currently sleeping and you can assume that they don't move.

In order to get a good view over the flamingos, each of the binoculars can be independently rotated to face any angle (not necessarily integer). Then, the binocular can be used to observe all flamingos that is located at the straight line passing through the binocular at the angle it is set. In other words, you can assign each binocular a direction corresponding to any straight line passing through the binocular, and the binocular will be able to see all flamingos located on that line.

Today, some kids from the prestigious Codeforces kindergarten went on a Field Study to the Zoo. Their teacher would like to set each binocular an angle to maximize the number of flamingos that can be seen by the binocular. The teacher is very interested in the sum of these values over all binoculars. Please help him find this sum.

## Input Format
The first line contains two space-separated integers n and m (1 ≤ n ≤ 106, 1 ≤ m ≤ 250), denoting the number of binoculars and the number of flamingos, respectively.

Then m lines follow, the i-th line will contain two space-separated integers xi and yi (1 ≤ xi, yi ≤ 109), which means that the i-th flamingo is located at point (xi, yi).

All flamingos will be located at distinct points.

## Output Format
Print a single integer denoting the maximum total number of flamingos that can be seen by all the binoculars.

## Examples
```input
5 5
2 1
4 1
3 2
4 3
4 4
```
```output
11
```

## Note
This picture shows the answer to the example test case.

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 are given a directed graph G with n vertices and m arcs (multiple arcs and self-loops are allowed). You have to paint each vertex of the graph into one of the k (k ≤ n) colors in such way that for all arcs of the graph leading from a vertex u to vertex v, vertex v is painted with the next color of the color used to paint vertex u.

The colors are numbered cyclically 1 through k. This means that for each color i (i < k) its next color is color i + 1. In addition, the next color of color k is color 1. Note, that if k = 1, then the next color for color 1 is again color 1.

Your task is to find and print the largest possible value of k (k ≤ n) such that it's possible to color G as described above with k colors. Note that you don't necessarily use all the k colors (that is, for each color i there does not necessarily exist a vertex that is colored with color i).

## Input Format
The first line contains two space-separated integers n and m (1 ≤ n, m ≤ 105), denoting the number of vertices and the number of arcs of the given digraph, respectively.

Then m lines follow, each line will contain two space-separated integers ai and bi (1 ≤ ai, bi ≤ n), which means that the i-th arc goes from vertex ai to vertex bi.

Multiple arcs and self-loops are allowed.

## Output Format
Print a single integer — the maximum possible number of the colors that can be used to paint the digraph (i.e. k, as described in the problem statement). Note that the desired value of k must satisfy the inequality 1 ≤ k ≤ n.

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

## Note
For the first example, with k = 2, this picture depicts the two colors (arrows denote the next color of that color).

With k = 2 a possible way to paint the graph is as follows.

It can be proven that no larger value for k exists for this test case.

For the second example, here's the picture of the k = 5 colors.

A possible coloring of the graph is:

For the third example, here's the picture of the k = 3 colors.

A possible coloring of the graph 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 c++17.

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:
```cpp
<your code here>
```

Execution time limit: 2.0 seconds
Memory limit: 256.0 MB

# Problem
A palindrome is a string that reads the same from left to right as from right to left. For example, abacaba, aaaa, abba, racecar are palindromes.

You are given a string $$$s$$$ consisting of lowercase Latin letters. The string $$$s$$$ is a palindrome.

You have to check whether it is possible to rearrange the letters in it to get another palindrome (not equal to the given string $$$s$$$).

## 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 lowercase Latin letters. This string is a palindrome.

## Output Format
For each test case, print YES if it is possible to rearrange the letters in the given string to get another palindrome. Otherwise, print NO.

You may print each letter in any case (YES, yes, Yes will all be recognized as positive answer, NO, no and nO will all be recognized as negative answer).

## Examples
```input
3
codedoc
gg
aabaa
```
```output
YES
NO
NO
```

## Note
In the first test case, it is possible to rearrange the letters in the palindrome codedoc to obtain the string ocdedco, which is different from the given string, but also a palindrome.

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 c++17.

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:
```cpp
<your code here>
```

Execution time limit: 2.0 seconds
Memory limit: 256.0 MB

# Problem
The only difference between easy and hard versions is the maximum values of $$$n$$$ and $$$q$$$.

You are given an array, consisting of $$$n$$$ integers. Initially, all elements are red.

You can apply the following operation to the array multiple times. During the $$$i$$$-th operation, you select an element of the array; then:

- if the element is red, it increases by $$$i$$$ and becomes blue;
- if the element is blue, it decreases by $$$i$$$ and becomes red.

The operations are numbered from $$$1$$$, i. e. during the first operation some element is changed by $$$1$$$ and so on.

You are asked $$$q$$$ queries of the following form:

- given an integer $$$k$$$, what can the largest minimum in the array be if you apply exactly $$$k$$$ operations to it?

Note that the operations don't affect the array between queries, all queries are asked on the initial array $$$a$$$.

## Input Format
The first line contains two integers $$$n$$$ and $$$q$$$ ($$$1 \le n, q \le 1000$$$) — the number of elements in the array and the number of queries.

The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 10^9$$$).

The third line contains $$$q$$$ integers $$$k_1, k_2, \dots, k_q$$$ ($$$1 \le k_j \le 10^9$$$).

## Output Format
For each query, print a single integer — the largest minimum that the array can have after you apply exactly $$$k$$$ operations to it.

## Examples
```input
4 10
5 2 8 4
1 2 3 4 5 6 7 8 9 10
```
```output
3 4 5 6 7 8 8 10 8 12
```
-----
```input
5 10
5 2 8 4 4
1 2 3 4 5 6 7 8 9 10
```
```output
3 4 5 6 7 8 9 8 11 8
```
-----
```input
2 5
2 3
10 6 8 1 3
```
```output
10 7 8 3 3
```

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 c++17.

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:
```cpp
<your code here>
```

Execution time limit: 2.0 seconds
Memory limit: 256.0 MB

# Problem
The only difference between easy and hard versions is the maximum values of $$$n$$$ and $$$q$$$.

You are given an array, consisting of $$$n$$$ integers. Initially, all elements are red.

You can apply the following operation to the array multiple times. During the $$$i$$$-th operation, you select an element of the array; then:

- if the element is red, it increases by $$$i$$$ and becomes blue;
- if the element is blue, it decreases by $$$i$$$ and becomes red.

The operations are numbered from $$$1$$$, i. e. during the first operation some element is changed by $$$1$$$ and so on.

You are asked $$$q$$$ queries of the following form:

- given an integer $$$k$$$, what can the largest minimum in the array be if you apply exactly $$$k$$$ operations to it?

Note that the operations don't affect the array between queries, all queries are asked on the initial array $$$a$$$.

## Input Format
The first line contains two integers $$$n$$$ and $$$q$$$ ($$$1 \le n, q \le 2 \cdot 10^5$$$) — the number of elements in the array and the number of queries.

The second line contains $$$n$$$ integers $$$a_1, a_2, \dots, a_n$$$ ($$$1 \le a_i \le 10^9$$$).

The third line contains $$$q$$$ integers $$$k_1, k_2, \dots, k_q$$$ ($$$1 \le k_j \le 10^9$$$).

## Output Format
For each query, print a single integer — the largest minimum that the array can have after you apply exactly $$$k$$$ operations to it.

## Examples
```input
4 10
5 2 8 4
1 2 3 4 5 6 7 8 9 10
```
```output
3 4 5 6 7 8 8 10 8 12
```
-----
```input
5 10
5 2 8 4 4
1 2 3 4 5 6 7 8 9 10
```
```output
3 4 5 6 7 8 9 8 11 8
```
-----
```input
2 5
2 3
10 6 8 1 3
```
```output
10 7 8 3 3
```

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 c++17.

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:
```cpp
<your code here>
```

Execution time limit: 4.0 seconds
Memory limit: 1024.0 MB

# Problem
Recall that the binomial coefficient $$$\binom{x}{y}$$$ is calculated as follows ($$$x$$$ and $$$y$$$ are non-negative integers):

- if $$$x < y$$$, then $$$\binom{x}{y} = 0$$$;
- otherwise, $$$\binom{x}{y} = \frac{x!}{y! \cdot (x-y)!}$$$.

You are given an array $$$a_1, a_2, \dots, a_n$$$ and an integer $$$k$$$. You have to calculate a new array $$$b_1, b_2, \dots, b_n$$$, where

- $$$b_1 = (\binom{1}{k} \cdot a_1) \bmod 998244353$$$;
- $$$b_2 = (\binom{2}{k} \cdot a_1 + \binom{1}{k} \cdot a_2) \bmod 998244353$$$;
- $$$b_3 = (\binom{3}{k} \cdot a_1 + \binom{2}{k} \cdot a_2 + \binom{1}{k} \cdot a_3) \bmod 998244353$$$, and so on.

Formally, $$$b_i = (\sum\limits_{j=1}^{i} \binom{i - j + 1}{k} \cdot a_j) \bmod 998244353$$$.

Note that the array is given in a modified way, and you have to output it in a modified way as well.

## Input Format
The only line of the input contains six integers $$$n$$$, $$$a_1$$$, $$$x$$$, $$$y$$$, $$$m$$$ and $$$k$$$ ($$$1 \le n \le 10^7$$$; $$$0 \le a_1, x, y < m$$$; $$$2 \le m \le 998244353$$$; $$$1 \le k \le 5$$$).

The array $$$[a_1, a_2, \dots, a_n]$$$ is generated as follows:

- $$$a_1$$$ is given in the input;
- for $$$2 \le i \le n$$$, $$$a_i = (a_{i-1} \cdot x + y) \bmod m$$$.

## Output Format
Since outputting up to $$$10^7$$$ integers might be too slow, you have to do the following:

Let $$$c_i = b_i \cdot i$$$ (without taking modulo $$$998244353$$$ after the multiplication). Print the integer $$$c_1 \oplus c_2 \oplus \dots \oplus c_n$$$, where $$$\oplus$$$ denotes the bitwise XOR operator.

## Examples
```input
5 8 2 3 100 2
```
```output
1283
```

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
There are $$$n$$$ nails driven into the wall, the $$$i$$$-th nail is driven $$$a_i$$$ meters above the ground, one end of the $$$b_i$$$ meters long rope is tied to it. All nails hang at different heights one above the other. One candy is tied to all ropes at once. Candy is tied to end of a rope that is not tied to a nail.

To take the candy, you need to lower it to the ground. To do this, Rudolph can cut some ropes, one at a time. Help Rudolph find the minimum number of ropes that must be cut to get the candy.

The figure shows an example of the first test:

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

The first line of each test case contains one integer $$$n$$$ ($$$1 \le n \le 50$$$) — the number of nails.

The $$$i$$$-th of the next $$$n$$$ lines contains two integers $$$a_i$$$ and $$$b_i$$$ ($$$1 \le a_i, b_i \le 200$$$) — the height of the $$$i$$$-th nail and the length of the rope tied to it, all $$$a_i$$$ are different.

It is guaranteed that the data is not contradictory, it is possible to build a configuration described in the statement.

## Output Format
For each test case print one integer — the minimum number of ropes that need to be cut to make the candy fall to the ground.

## Examples
```input
4
3
4 3
3 1
1 2
4
9 2
5 2
7 7
3 4
5
11 7
5 10
12 9
3 2
1 5
3
5 6
4 5
7 7
```
```output
2
2
3
0
```

Now solve the problem and return the code.