Sure! Let's create a test case generator for the problem statement you provided.

### Test Case Generator for the Problem Statement

The goal is to generate random strings \( S \) and \( T \) of equal length such that they follow the constraints given. The strings should consist of lowercase English letters and have lengths between 1 and 200,000.

Here is the code for the test case generator:

```python
import random
import string

def tcgen():
    # Define the length of the strings
    length = random.randint(1, 2 * 10**5)
    
    # Generate two random strings of the same length
    S = ''.join(random.choices(string.ascii_lowercase, k=length))
    T = ''.join(random.choices(string.ascii_lowercase, k=length))
    
    return S, T

# Example usage:
for _ in range(5):
    S, T = tcgen()
    print(f"S: {S}")
    print(f"T: {T}")
    print()
```

Here is a breakdown of the code:
1. **Import the necessary modules**: `random` for generating random numbers and `string` for accessing lowercase English letters.
2. **Define the length of the strings**: A random integer between 1 and 200,000 is chosen.
3. **Generate two random strings**: Both strings \( S \) and \( T \) are generated using `random.choices` which selects random characters from `string.ascii_lowercase` and forms a string of the specified length.
4. **Return the generated strings**: The function returns the two generated strings \( S \) and \( T \).

This function will generate random test cases for the problem statement you provided. Note that the generated strings are completely random and do not guarantee any specific relationship between \( S \) and \( T \), which is suitable for general testing.