Problem p03365 - Generation 1

Orig Description

Score : 800 points
Problem StatementThere are N squares lining up in a row, numbered 1 through N from left to right.
Initially, all squares are white.
We also have N-1 painting machines, numbered 1 through N-1.
When operated, Machine i paints Square i and i+1 black.
Snuke will operate these machines one by one.
The order in which he operates them is represented by a permutation of (1, 2, ..., N-1), P, which means that the i-th operated machine is Machine P_i.
Here, the score of a permutation P is defined as the number of machines that are operated before all the squares are painted black for the first time, when the machines are operated in the order specified by P.
Snuke has not decided what permutation P to use, but he is interested in the scores of possible permutations.
Find the sum of the scores over all possible permutations for him.
Since this can be extremely large, compute the sum modulo 10^9+7.
Constraints
2 \leq N \leq 10^6
InputInput is given from Standard Input in the following format:
N
OutputPrint the sum of the scores over all possible permutations, modulo 10^9+7.
Sample Input 14
Sample Output 116
There are six possible permutations as P.
Among them, P = (1, 3, 2) and P = (3, 1, 2) have a score of 2, and the others have a score of 3.
Thus, the answer is 2 \times 2 + 3 \times 4 = 16.
Sample Input 22
Sample Output 21
There is only one possible permutation: P = (1), which has a score of 1.
Sample Input 35
Sample Output 384
Sample Input 4100000
Sample Output 4341429644

Extracted Specification

An integer N (2 ≤ N ≤ 10^6), representing some quantity or size.

### Example Input:

```
4
```

### Function Signature:
Write a function f(N) that takes in the input.
def f(N: int):
    ''' 
    N: an integer 
    '''

Test Case Generator

Certainly! Below is a Python function that generates test cases for the given problem statement:

```python
import random

def tcgen():
    # Define the constraints
    MIN_N = 2
    MAX_N = 10**6
    MOD = 10**9 + 7
    
    # Generate a random value for N within the given constraints
    N = random.randint(MIN_N, MAX_N)
    
    return N

# Example usage:
if __name__ == "__main__":
    # Generate a test case
    print(tcgen())
```

### Explanation
1. **Constraints**:
    - \( N \) ranges from 2 to \( 10^6 \).
   
2. **Function `tcgen`**:
    - Randomly selects an integer \( N \) within the given constraints.
    - Returns \( N \).

### Example Usage
Run the script and it will print a random integer \( N \) within the specified range, which represents the number of squares.

### Testing
To test the function, you can simply call `tcgen()` multiple times to see different generated values for \( N \). Make sure to integrate this with your solution to verify correctness with various input sizes.

```python
if __name__ == "__main__":
    for _ in range(5):  # Generate 5 test cases for demonstration
        print(tcgen())
```

This would give you a good spread of values for \( N \) to test against your solution for correctness and performance.

Extract Arguments

def extract_arguments(fh):
    N = int(fh.readline().strip())
    return (N,)