Certainly! Here is a test case generator for the given problem statement:

```python
import random

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

# Example usage:
N, K, a = tcgen()
print(N, K)
print('\n'.join(map(str, a)))
```

This script generates random test cases according to the problem constraints:

1. `N` is an integer between 1 and \(2 \times 10^5\).
2. `K` is an integer between 1 and \(10^9\).
3. The sequence `a` contains `N` integers, each between 1 and \(10^9\).

The example usage at the end demonstrates how to generate and print a test case. Each test case is generated randomly, ensuring a wide range of possible scenarios to test the solution.