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 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: 512.0 MB

# Problem
Kostya likes Codeforces contests very much. However, he is very disappointed that his solutions are frequently hacked. That's why he decided to obfuscate (intentionally make less readable) his code before upcoming contest.

To obfuscate the code, Kostya first looks at the first variable name used in his program and replaces all its occurrences with a single symbol a, then he looks at the second variable name that has not been replaced yet, and replaces all its occurrences with b, and so on. Kostya is well-mannered, so he doesn't use any one-letter names before obfuscation. Moreover, there are at most 26 unique identifiers in his programs.

You are given a list of identifiers of some program with removed spaces and line breaks. Check if this program can be a result of Kostya's obfuscation.

## Input Format
In the only line of input there is a string S of lowercase English letters (1 ≤ |S| ≤ 500) — the identifiers of a program with removed whitespace characters.

## Output Format
If this program can be a result of Kostya's obfuscation, print "YES" (without quotes), otherwise print "NO".

## Examples
```input
abacaba
```
```output
YES
```
-----
```input
jinotega
```
```output
NO
```

## Note
In the first sample case, one possible list of identifiers would be "number string number character number string number". Here how Kostya would obfuscate the program:

- replace all occurences of number with a, the result would be "a string a character a string a",
- replace all occurences of string with b, the result would be "a b a character a b a",
- replace all occurences of character with c, the result would be "a b a c a b a",
- all identifiers have been replaced, thus the obfuscation is finished.

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: 512.0 MB

# Problem
Misha and Vanya have played several table tennis sets. Each set consists of several serves, each serve is won by one of the players, he receives one point and the loser receives nothing. Once one of the players scores exactly k points, the score is reset and a new set begins.

Across all the sets Misha scored a points in total, and Vanya scored b points. Given this information, determine the maximum number of sets they could have played, or that the situation is impossible.

Note that the game consisted of several complete sets.

## Input Format
The first line contains three space-separated integers k, a and b (1 ≤ k ≤ 109, 0 ≤ a, b ≤ 109, a + b > 0).

## Output Format
If the situation is impossible, print a single number -1. Otherwise, print the maximum possible number of sets.

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

## Note
Note that the rules of the game in this problem differ from the real table tennis game, for example, the rule of "balance" (the winning player has to be at least two points ahead to win a set) has no power within the present problem.

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: 512.0 MB

# Problem
Artsem has a friend Saunders from University of Chicago. Saunders presented him with the following problem.

Let [n] denote the set {1, ..., n}. We will also write f: [x] → [y] when a function f is defined in integer points 1, ..., x, and all its values are integers from 1 to y.

Now then, you are given a function f: [n] → [n]. Your task is to find a positive integer m, and two functions g: [n] → [m], h: [m] → [n], such that g(h(x)) = x for all $$x \in [m]$$, and h(g(x)) = f(x) for all $$x \in [n]$$, or determine that finding these is impossible.

## Input Format
The first line contains an integer n (1 ≤ n ≤ 105).

The second line contains n space-separated integers — values f(1), ..., f(n) (1 ≤ f(i) ≤ n).

## Output Format
If there is no answer, print one integer -1.

Otherwise, on the first line print the number m (1 ≤ m ≤ 106). On the second line print n numbers g(1), ..., g(n). On the third line print m numbers h(1), ..., h(m).

If there are several correct answers, you may output any of them. It is guaranteed that if a valid answer exists, then there is an answer satisfying the above restrictions.

## Examples
```input
3
1 2 3
```
```output
3
1 2 3
1 2 3
```
-----
```input
3
2 2 2
```
```output
1
1 1 1
2
```
-----
```input
2
2 1
```
```output
-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 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: 512.0 MB

# Problem
Vanya wants to minimize a tree. He can perform the following operation multiple times: choose a vertex v, and two disjoint (except for v) paths of equal length a0 = v, a1, ..., ak, and b0 = v, b1, ..., bk. Additionally, vertices a1, ..., ak, b1, ..., bk must not have any neighbours in the tree other than adjacent vertices of corresponding paths. After that, one of the paths may be merged into the other, that is, the vertices b1, ..., bk can be effectively erased:

Help Vanya determine if it possible to make the tree into a path via a sequence of described operations, and if the answer is positive, also determine the shortest length of such path.

## Input Format
The first line of input contains the number of vertices n (2 ≤ n ≤ 2·105).

Next n - 1 lines describe edges of the tree. Each of these lines contains two space-separated integers u and v (1 ≤ u, v ≤ n, u ≠ v) — indices of endpoints of the corresponding edge. It is guaranteed that the given graph is a tree.

## Output Format
If it is impossible to obtain a path, print -1. Otherwise, print the minimum number of edges in a possible path.

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

## Note
In the first sample case, a path of three edges is obtained after merging paths 2 - 1 - 6 and 2 - 4 - 5.

It is impossible to perform any operation in the second sample case. For example, it is impossible to merge paths 1 - 3 - 4 and 1 - 5 - 6, since vertex 6 additionally has a neighbour 7 that is not present in the corresponding path.

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: 5.0 seconds
Memory limit: 512.0 MB

# Problem
If you have gone that far, you'll probably skip unnecessary legends anyway...

You are given a binary string $$s = s_0 \ldots s_{m-1}$$ and an integer $$N = p_{1}^{\alpha_{1}} \ldots p_{n}^{\alpha_{n}}$$. Find the number of integers k, 0 ≤ k < N, such that for all i = 0, 1, ..., m - 1

$$\gcd(k+i,N)=1 \text{ if and only if } s_i=1$$

## Input Format
In the first line of input there is a string s consisting of 0's and 1's (1 ≤ |s| ≤ 40).

In the next line of input there is an integer n (1 ≤ n ≤ 5·105).

Each of the next n lines contains two space-separated integers pi, αi (1 ≤ pi, αi ≤ 109, pi is prime). All pi are distinct.

## Output Format
A single integer — the answer to the problem.

## Examples
```input
1
2
2 1
3 1
```
```output
2
```
-----
```input
01
2
3 2
5 1
```
```output
15
```
-----
```input
1011
1
3 1000000000
```
```output
411979884
```

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
Igor the analyst has adopted n little bunnies. As we all know, bunnies love carrots. Thus, Igor has bought a carrot to be shared between his bunnies. Igor wants to treat all the bunnies equally, and thus he wants to cut the carrot into n pieces of equal area.

Formally, the carrot can be viewed as an isosceles triangle with base length equal to 1 and height equal to h. Igor wants to make n - 1 cuts parallel to the base to cut the carrot into n pieces. He wants to make sure that all n pieces have the same area. Can you help Igor determine where to cut the carrot so that each piece have equal area?

Illustration to the first example.

## Input Format
The first and only line of input contains two space-separated integers, n and h (2 ≤ n ≤ 1000, 1 ≤ h ≤ 105).

## Output Format
The output should contain n - 1 real numbers x1, x2, ..., xn - 1. The number xi denotes that the i-th cut must be made xi units away from the apex of the carrot. In addition, 0 < x1 < x2 < ... < xn - 1 < h must hold.

Your output will be considered correct if absolute or relative error of every number in your output doesn't exceed 10 - 6.

Formally, let your answer be a, and the jury's answer be b. Your answer is considered correct if $$\frac{|a-b|}{\max(1,b)} \leq 10^{-6}$$.

## Examples
```input
3 2
```
```output
1.154700538379 1.632993161855
```
-----
```input
2 100000
```
```output
70710.678118654752
```

## Note
Definition of isosceles triangle: https://en.wikipedia.org/wiki/Isosceles_triangle.

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 found a mysterious function f. The function takes two strings s1 and s2. These strings must consist only of lowercase English letters, and must be the same length.

The output of the function f is another string of the same length. The i-th character of the output is equal to the minimum of the i-th character of s1 and the i-th character of s2.

For example, f("ab", "ba") = "aa", and f("nzwzl", "zizez") = "niwel".

You found two strings x and y of the same length and consisting of only lowercase English letters. Find any string z such that f(x, z) = y, or print -1 if no such string z exists.

## Input Format
The first line of input contains the string x.

The second line of input contains the string y.

Both x and y consist only of lowercase English letters, x and y have same length and this length is between 1 and 100.

## Output Format
If there is no string z such that f(x, z) = y, print -1.

Otherwise, print a string z such that f(x, z) = y. If there are multiple possible answers, print any of them. The string z should be the same length as x and y and consist only of lowercase English letters.

## Examples
```input
ab
aa
```
```output
ba
```
-----
```input
nzwzl
niwel
```
```output
xiyez
```
-----
```input
ab
ba
```
```output
-1
```

## Note
The first case is from the statement.

Another solution for the second case is "zizez"

There is no solution for the third case. That is, there is no z such that f("ab", z) =  "ba".

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: 1.0 seconds
Memory limit: 256.0 MB

# Problem
Captain Bill the Hummingbird and his crew recieved an interesting challenge offer. Some stranger gave them a map, potion of teleportation and said that only this potion might help them to reach the treasure.

Bottle with potion has two values x and y written on it. These values define four moves which can be performed using the potion:

- $$(a,b)\rightarrow(a+x,b+y)$$
- $$(a,b)\rightarrow(a+x,b-y)$$
- $$(a,b)\rightarrow(a-x,b+y)$$
- $$(a,b)\rightarrow(a-x,b-y)$$

Map shows that the position of Captain Bill the Hummingbird is (x1, y1) and the position of the treasure is (x2, y2).

You task is to tell Captain Bill the Hummingbird whether he should accept this challenge or decline. If it is possible for Captain to reach the treasure using the potion then output "YES", otherwise "NO" (without quotes).

The potion can be used infinite amount of times.

## Input Format
The first line contains four integer numbers x1, y1, x2, y2 ( - 105 ≤ x1, y1, x2, y2 ≤ 105) — positions of Captain Bill the Hummingbird and treasure respectively.

The second line contains two integer numbers x, y (1 ≤ x, y ≤ 105) — values on the potion bottle.

## Output Format
Print "YES" if it is possible for Captain to reach the treasure using the potion, otherwise print "NO" (without quotes).

## Examples
```input
0 0 0 6
2 3
```
```output
YES
```
-----
```input
1 1 3 6
1 5
```
```output
NO
```

## Note
In the first example there exists such sequence of moves:

1. $$(0,0)\rightarrow(2,3)$$ — the first type of move
2. $$(2,3)\rightarrow(0,6)$$ — the third type of move

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
After returning from the army Makes received a gift — an array a consisting of n positive integer numbers. He hadn't been solving problems for a long time, so he became interested to answer a particular question: how many triples of indices (i,  j,  k) (i < j < k), such that ai·aj·ak is minimum possible, are there in the array? Help him with it!

## Input Format
The first line of input contains a positive integer number n (3 ≤ n ≤ 105) — the number of elements in array a. The second line contains n positive integer numbers ai (1 ≤ ai ≤ 109) — the elements of a given array.

## Output Format
Print one number — the quantity of triples (i,  j,  k) such that i,  j and k are pairwise distinct and ai·aj·ak is minimum possible.

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

## Note
In the first example Makes always chooses three ones out of four, and the number of ways to choose them is 4.

In the second example a triple of numbers (1, 2, 3) is chosen (numbers, not indices). Since there are two ways to choose an element 3, then the answer is 2.

In the third example a triple of numbers (1, 1, 2) is chosen, and there's only one way to choose indices.

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: 1.0 seconds
Memory limit: 256.0 MB

# Problem
Ivan likes to learn different things about numbers, but he is especially interested in really big numbers. Ivan thinks that a positive integer number x is really big if the difference between x and the sum of its digits (in decimal representation) is not less than s. To prove that these numbers may have different special properties, he wants to know how rare (or not rare) they are — in fact, he needs to calculate the quantity of really big numbers that are not greater than n.

Ivan tried to do the calculations himself, but soon realized that it's too difficult for him. So he asked you to help him in calculations.

## Input Format
The first (and the only) line contains two integers n and s (1 ≤ n, s ≤ 1018).

## Output Format
Print one integer — the quantity of really big numbers that are not greater than n.

## Examples
```input
12 1
```
```output
3
```
-----
```input
25 20
```
```output
0
```
-----
```input
10 9
```
```output
1
```

## Note
In the first example numbers 10, 11 and 12 are really big.

In the second example there are no really big numbers that are not greater than 25 (in fact, the first really big number is 30: 30 - 3 ≥ 20).

In the third example 10 is the only really big number (10 - 1 ≥ 9).

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
After studying the beacons Mister B decided to visit alien's planet, because he learned that they live in a system of flickering star Moon. Moreover, Mister B learned that the star shines once in exactly T seconds. The problem is that the star is yet to be discovered by scientists.

There are n astronomers numerated from 1 to n trying to detect the star. They try to detect the star by sending requests to record the sky for 1 second.

The astronomers send requests in cycle: the i-th astronomer sends a request exactly ai second after the (i - 1)-th (i.e. if the previous request was sent at moment t, then the next request is sent at moment t + ai); the 1-st astronomer sends requests a1 seconds later than the n-th. The first astronomer sends his first request at moment 0.

Mister B doesn't know the first moment the star is going to shine, but it's obvious that all moments at which the star will shine are determined by the time of its shine moment in the interval [0, T). Moreover, this interval can be split into T parts of 1 second length each of form [t, t + 1), where t = 0, 1, 2, ..., (T - 1).

Mister B wants to know how lucky each astronomer can be in discovering the star first.

For each astronomer compute how many segments of form [t, t + 1) (t = 0, 1, 2, ..., (T - 1)) there are in the interval [0, T) so that this astronomer is the first to discover the star if the first shine of the star happens in this time interval.

## Input Format
The first line contains two integers T and n (1 ≤ T ≤ 109, 2 ≤ n ≤ 2·105).

The second line contains n integers a1, a2, ..., an (1 ≤ ai ≤ 109).

## Output Format
Print n integers: for each astronomer print the number of time segments describer earlier.

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

## Note
In the first sample test the first astronomer will send requests at moments t1 = 0, 5, 10, ..., the second — at moments t2 = 3, 8, 13, .... That's why interval [0, 1) the first astronomer will discover first at moment t1 = 0, [1, 2) — the first astronomer at moment t1 = 5, [2, 3) — the first astronomer at moment t1 = 10, and [3, 4) — the second astronomer at moment t2 = 3.

In the second sample test interval [0, 1) — the first astronomer will discover first, [1, 2) — the second astronomer, [2, 3) — the third astronomer, [3, 4) — the fourth astronomer, [4, 5) — the first astronomer.

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 order to fly to the Moon Mister B just needs to solve the following problem.

There is a complete indirected graph with n vertices. You need to cover it with several simple cycles of length 3 and 4 so that each edge is in exactly 2 cycles.

We are sure that Mister B will solve the problem soon and will fly to the Moon. Will you?

## Input Format
The only line contains single integer n (3 ≤ n ≤ 300).

## Output Format
If there is no answer, print -1.

Otherwise, in the first line print k (1 ≤ k ≤ n2) — the number of cycles in your solution.

In each of the next k lines print description of one cycle in the following format: first print integer m (3 ≤ m ≤ 4) — the length of the cycle, then print m integers v1, v2, ..., vm (1 ≤ vi ≤ n) — the vertices in the cycle in the traverse order. Each edge should be in exactly two cycles.

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

Now solve the problem and return the code.