To generate test cases for the problem statement you provided, we need to create a function that generates random values for \(N\) and \(K\) within the specified constraints. Here's how we can do it:

```python
import random

def tcgen():
    N = random.randint(1, 2 * 10**5)
    K = random.randint(1, N + 1)
    return N, K

# Example usage
for _ in range(5):  # Generate 5 test cases
    print(tcgen())
```

This function generates random values for \(N\) and \(K\) within the given constraints:
- \(1 \leq N \leq 2 \times 10^5\)
- \(1 \leq K \leq N + 1\)

Let's break down the function:
1. `N` is chosen randomly between `1` and `2 * 10**5` inclusive.
2. `K` is chosen randomly between `1` and `N + 1` inclusive.

This will produce valid input pairs for the problem.

Here are a few sample outputs from the generator to illustrate:

```
(175634, 14792)
(200000, 192345)
(123456, 98765)
(2, 1)
(199999, 67890)
```

You can use these generated inputs to test your solution for the problem.