Certainly! Below is a test case generator for the given problem statement.

```python
import random

def generate_test_case():
    # Constraints
    N = random.randint(1, 200)
    points = set()
    
    while len(points) < N:
        x = random.randint(0, 9999)
        y = random.randint(0, 9999)
        # Ensure all points are unique
        if (x, y) not in points:
            points.add((x, y))
    
    points_list = list(points)
    
    return N, points_list

def format_test_case(N, points_list):
    formatted_input = f"{N}\n"
    for x, y in points_list:
        formatted_input += f"{x} {y}\n"
        
    return formatted_input

# Example Usage
N, points_list = generate_test_case()
test_case_input = format_test_case(N, points_list)
print(test_case_input)
```

This generator creates a test case for the given problem by generating a random number \(N\) of unique points within the specified range [0, 9999] for both \(x\) and \(y\) coordinates.

### Steps to Generate a Test Case:

1. **Generate a Random Number of Points**: `N` is chosen randomly between 1 and 200.
2. **Generate Unique Points**: Use a set to ensure the generated points are unique.
3. **Format the Input**: Convert the list of points into the required input format for the problem.

### Example Usage

Running the generator will produce a test case in the format expected by the problem. The `format_test_case` function converts the generated points into a string format that mimics standard input.

### Example Output

Here's an example of what the output might look like:

```
5
1234 5678
2345 6789
3456 7890
4567 8901
5678 9012
```

This output can be directly used as input for your problem, ensuring it adheres to the constraints and requirements given in the problem statement.