Problem p03500 - Generation 3

Orig Description

Score : 1200 points
Problem StatementThere are N non-negative integers written on the blackboard: A_1, ..., A_N.
Snuke can perform the following two operations at most K times in total in any order:
Operation A: Replace each integer X on the blackboard with X divided by 2, rounded down to the nearest integer.
Operation B: Replace each integer X on the blackboard with X minus 1. This operation cannot be performed if one or more 0s are written on the blackboard.
Find the number of the different possible combinations of integers written on the blackboard after Snuke performs the operations, modulo 1,000,000,007.
Constraints
1 \leq N \leq 200
1 \leq A_i \leq 10^{18}
1 \leq K \leq 10^{18}
A_i and K are integers.
InputInput is given from Standard Input in the following format:
N K
A_1 A_2 ... A_N
OutputPrint the number of the different possible combinations of integers written on the blackboard after Snuke performs the operations, modulo 1,000,000,007.
Sample Input 12 2
5 7
Sample Output 16
There are six possible combinations of integers on the blackboard: (1, 1), (1, 2), (2, 3), (3, 5), (4, 6) and (5, 7).
For example, (1, 2) can be obtained by performing Operation A and Operation B in this order.
Sample Input 23 4
10 13 22
Sample Output 220
Sample Input 31 100
10
Sample Output 311
Sample Input 410 123456789012345678
228344079825412349 478465001534875275 398048921164061989 329102208281783917 779698519704384319 617456682030809556 561259383338846380 254083246422083141 458181156833851984 502254767369499613
Sample Output 4164286011

Extracted Specification

An integer N (1 ≤ N ≤ 200), representing some quantity or size.
An integer K (1 ≤ K ≤ 10^18), representing some value or limit.
A list of N integers A, where each integer is between 1 and 10^18.

### Example Input:

```
2 2
5 7
```

### Function Signature:
Write a function f(N, K, A) that takes in the input.
def f(N: int, K: int, A: List[int]):
    '''
    N: an integer
    K: an integer
    A: a list of integers
    '''

Test Case Generator

Certainly! Below is a Python function `tcgen` that generates test cases for the provided competitive programming problem.

```python
import random

def tcgen():
    # Setting the constraints
    N = random.randint(1, 200)
    
    A = [random.randint(1, 10**18) for _ in range(N)]
    
    K = random.randint(1, 10**18)
    
    return N, K, A

# Example usage:
N, K, A = tcgen()
print(N)
print(K)
print(" ".join(map(str, A)))
```

### Explanation:
- **N**: This is set to a random integer between 1 and 200 inclusive, which is the constraint for the number of integers written on the blackboard.
- **A**: This is a list of integers where each element is a random integer between 1 and \(10^{18}\), adhering to the constraints for the integers on the blackboard.
- **K**: This is set to a random integer between 1 and \(10^{18}\), following the constraint for the maximum number of operations.

### How to use:
1. Call the `tcgen` function to generate a test case.
2. The function will print:
   - The number of integers \(N\) on the blackboard.
   - The number of operations \(K\).
   - The list \(A\) of \(N\) integers.

You can use the printed values as input for the problem to test your solution or any other use case you need.

Extract Arguments

def extract_arguments(fh):
    N, K = map(int, fh.readline().strip().split())
    A = list(map(int, fh.readline().strip().split()))
    return N, K, A

# if __name__ == "__main__":
#     input_path = sys.argv[1]
#     with open(input_path, 'r') as fh: 
#     N, K, A = extract_arguments(fh)
#     f(N, K, A)