Sure, I'll write a test case generator for the given problem statement. The problem involves generating constraints and values of \( A_i \) and \( B_i \) for \( N \) integers, where \( 2 \leq N \leq 2 \times 10^5 \) and \( 1 \leq A_i \leq B_i \leq 10^9 \).

Here's the test case generator in Python:

```python
import random

def tcgen():
    # Generate a random N in the valid range
    N = random.randint(2, 2 * 10**5)
    
    # Generate A and B arrays with valid constraints
    A = []
    B = []
    for _ in range(N):
        a = random.randint(1, 10**9)
        b = random.randint(a, 10**9)
        A.append(a)
        B.append(b)
    
    # Create the formatted input as a string
    input_str = f"{N}\n"
    for a, b in zip(A, B):
        input_str += f"{a} {b}\n"

    return input_str

# Example usage
print(tcgen())
```

This test case generator will create large and valid test cases according to the given constraints. The function `tcgen()` generates a random integer \( N \) within the specified range, and then for each \( i \), it generates random values for \( A_i \) and \( B_i \) such that \( 1 \leq A_i \leq B_i \leq 10^9 \). It then formats these values into a string that matches the expected input format for the problem.

You can call this function to generate different test cases, and you can use the output to test your solution to the competitive programming problem.