Sure, let's write a test case generator for the competitive programming problem you've described. The generator will create random values for \( N \), \( K \), and the set \( A \).

Here is the Python code for the test case generator:

```python
import random

def tcgen():
    # Generate a random value for N (1 ≤ N ≤ 100)
    N = random.randint(1, 100)
    
    # Generate a random value for K (1 ≤ K ≤ 10^5)
    K = random.randint(1, 10**5)
    
    # Generate a sorted list of N unique integers for A, where 1 ≤ a_1 < a_2 < ... < a_N ≤ K
    A = sorted(random.sample(range(1, K + 1), N))
    
    return N, K, A

# Generate a random test case and print it
if __name__ == "__main__":
    N, K, A = tcgen()
    print(N)
    print(K)
    print(" ".join(map(str, A)))
```

This script will generate a test case with the following steps:
1. Randomly select \( N \) within the range \([1, 100]\).
2. Randomly select \( K \) within the range \([1, 10^5]\).
3. Generate a sorted list \( A \) of \( N \) unique integers such that \( 1 \leq a_1 < a_2 < \ldots < a_N \leq K \).

You can run this script to get random test cases for the problem. Each run will produce different values for \( N \), \( K \), and the list \( A \), ensuring a wide variety of test cases for thorough testing.

Here is an example of how to use this generator:

```python
# Example usage
N, K, A = tcgen()
print(f"N = {N}")
print(f"K = {K}")
print(f"A = {A}")
```

This will print a generated test case in a format that you can use directly for your problem statement.