Certainly! Here is the test case generator function for the given problem statement:

```python
import random

def generate_test_case():
    N = random.randint(1, 10**5)
    a = [random.randint(1, 10**9) for _ in range(N)]
    
    return N, a

# Function to print the test case in the required format
def print_test_case():
    N, a = generate_test_case()
    print(N)
    for value in a:
        print(value)

# Example usage
if __name__ == "__main__":
    print_test_case()
```

Explanation of the code:
1. `N` is randomly generated within the range \(1 \leq N \leq 10^5\).
2. The list `a` is generated with `N` elements, where each element is a random integer between \(1 \leq a_i \leq 10^9\).
3. The function `print_test_case` prints `N` followed by each element in the list `a` to match the input format required by the problem statement.

This generator can be used to create random test cases for the described problem, allowing you to test the solution under various random conditions.