Here's a test case generator for the given problem statement:

```python
import random

def tcgen():
    # Generate a random integer K within the given constraints
    K = random.randint(1, 200)
    
    return K

# Example usage
if __name__ == "__main__":
    # Generate a test case
    K = tcgen()
    
    # Print the test case
    print(K)
```

This generator will produce a random integer \( K \) in the range from 1 to 200, as required by the problem constraints. You can then use this test case to run your solution and verify its correctness.

To explain it step by step:

1. **Importing the `random` module:** This is necessary to generate random numbers.
2. **Defining the `tcgen` function:** This function will generate a single random integer \( K \) within the range [1, 200].
3. **Using `random.randint(1, 200)` to generate the random integer \( K \):** This method ensures that \( K \) will always be within the specified constraints.
4. **Returning \( K \):** The function returns the generated \( K \).
5. **Usage example:** This is a simple example of how to use the `tcgen` function to generate and print a test case.

You can execute the provided code snippet to generate and print random test cases for your problem.