Sure, I can write a test case generator for this new problem statement. Here is the generator:

```python
import random

def tcgen():
    # Given constraints
    N = random.randint(1, 2 * 10**5)
    K = random.randint(1, 10**9)
    
    # Generate the sequence a of length N, with values between 1 and 10^9
    a = [random.randint(1, 10**9) for _ in range(N)]
    
    return N, K, a

# Example usage of the test case generator
N, K, a = tcgen()
print(N, K)
for ai in a:
    print(ai)
```

### Explanation:
1. **N**: The length of the sequence is randomly chosen between 1 and 200,000 inclusive.
2. **K**: The integer K is randomly chosen between 1 and 1,000,000,000 inclusive.
3. **a**: The sequence \(a\) consists of \(N\) integers, each randomly chosen between 1 and 1,000,000,000 inclusive.

This generator ensures that the constraints are respected and generates a valid input for the problem. To run the generator and print a test case, you can call `tcgen()` and print the values as shown in the example usage.