Sure! Based on the problem statement you provided, here is a test case generator for the problem:

```python
import random

def tcgen():
    # Constraints
    N = random.randint(1, 200000)
    M = random.randint(1, 200000)
    
    # Generating L and R values within the constraints
    L = []
    R = []
    for _ in range(N):
        Li = random.randint(0, M)
        Ri = random.randint(Li + 1, M + 1)
        L.append(Li)
        R.append(Ri)
    
    return N, M, L, R

# Example usage
N, M, L, R = tcgen()
print(N, M)
for Li, Ri in zip(L, R):
    print(Li, Ri)
```

This function generates a random test case based on the constraints provided in the problem statement. Here is what the function does step-by-step:

1. It randomly selects the number of people \(N\) and the number of chairs \(M\) within the given constraints.
2. It generates random pairs of coordinates \(L_i\) and \(R_i\) for each person, ensuring that \(0 \leq L_i < R_i \leq M + 1\).
3. It returns the generated values so they can be used to create a test case.

You can run the example usage to see a randomly generated test case printed in the required format.