Certainly! Here's a detailed test case generator for the "Symmetric Ternary" problem:

```python
import random

def generate_test_case():
    w = random.randint(1, 100000)
    return w

# Example usage:
if __name__ == "__main__":
    random.seed(42)  # For reproducibility
    for _ in range(5):  # Generate 5 example test cases
        test_case = generate_test_case()
        print(test_case)

```

This function `generate_test_case` generates a random integer `w` within the specified range (1 ≤ w ≤ 100000). You can run this script to generate multiple test cases. 

Here's a breakdown of how it works:
1. **`random.randint(1, 100000)`**: This will generate a random integer within the given range.
2. **Example Usage Section**: This part of the code is to illustrate how you can generate multiple test cases. By running the script, you will get 5 different test cases printed out.

You can modify the number of test cases to generate by changing the range in the `for` loop.

If you need to integrate this into a larger test suite or use it in a competitive programming environment, you can adapt the function accordingly.