Sure, I can help you write a test case generator for the given problem statement. Here's a Python function that generates test cases based on the constraints provided:

```python
import random

def tcgen():
    # Number of people and chairs
    N = random.randint(1, 2 * 10**5)
    M = random.randint(1, 2 * 10**5)
    
    # Generate L and R values for each person
    LR = []
    for _ in range(N):
        L = random.randint(0, M)
        R = random.randint(L + 1, M + 1)
        LR.append((L, R))
    
    return N, M, LR

def format_input(N, M, LR):
    input_str = f"{N} {M}\n"
    for L, R in LR:
        input_str += f"{L} {R}\n"
    return input_str.strip()

# Example usage
if __name__ == "__main__":
    N, M, LR = tcgen()
    test_case = format_input(N, M, LR)
    print(test_case)
```

This script will generate random test cases based on the constraints given in the problem statement. Here's a breakdown of the code:

1. **tcgen function**:
   - Randomly generates `N`, the number of people, and `M`, the number of chairs, within the specified constraints.
   - For each person, it generates random values for `L` and `R` such that `0 ≤ L < R ≤ M + 1`.
   - Collects all these `(L, R)` pairs in the list `LR`.

2. **format_input function**:
   - Takes the generated values (`N`, `M`, and `LR`) and formats them into the input format expected by the problem.
   - Returns the formatted string.

3. **Example usage**:
   - Calls the `tcgen` function to generate a test case.
   - Formats the generated test case using `format_input`.
   - Prints the test case.

You can run the script multiple times to generate different test cases, or you can modify it to generate a specific number of test cases and write them to files for further testing.