Sure, I can help you with that. Here's a test case generator for the "Grid" problem:

```python
import random

def tcgen():
    # Define the maximum values for r and c
    max_r = 1000
    max_c = 1000
    
    # Randomly select values for r and c within the specified range
    r = random.randint(1, max_r)
    c = random.randint(1, max_c)
    
    # Randomly select coordinates (a1, a2) and (b1, b2) within the grid
    a1 = random.randint(0, r - 1)
    a2 = random.randint(0, c - 1)
    b1 = random.randint(0, r - 1)
    b2 = random.randint(0, c - 1)
    
    return r, c, a1, a2, b1, b2

# Example usage
if __name__ == "__main__":
    for _ in range(10):  # Generate 10 test cases
        print(tcgen())
```

This script randomly generates test cases for the given problem within the specified constraints (1 ≤ r, c ≤ 1,000 and appropriate ranges for the coordinates). Here's how it works:

1. **Grid Size**: It randomly selects the number of rows \(r\) and columns \(c\) within the range [1, 1000].
2. **Coordinates**: It randomly selects the start coordinates \((a1, a2)\) and the end coordinates \((b1, b2)\) ensuring they are within the grid bounds.
3. **Output**: The function returns a tuple with the values \((r, c, a1, a2, b1, b2)\).

To generate multiple test cases, you can call `tcgen()` multiple times as shown in the example usage.