Here is a Python function to generate test cases for the described problem statement. This function will generate random values for \(X\), \(Y\), \(Z\), \(K\), and the deliciousness arrays \(A\), \(B\), and \(C\):

```python
import random

def tcgen():
    X = random.randint(1, 1000)
    Y = random.randint(1, 1000)
    Z = random.randint(1, 1000)
    K = random.randint(1, min(3000, X * Y * Z))
    
    A = [random.randint(1, 10**10) for _ in range(X)]
    B = [random.randint(1, 10**10) for _ in range(Y)]
    C = [random.randint(1, 10**10) for _ in range(Z)]
    
    return X, Y, Z, K, A, B, C

def format_tc(X, Y, Z, K, A, B, C):
    input_str = f"{X} {Y} {Z} {K}\n"
    input_str += " ".join(map(str, A)) + "\n"
    input_str += " ".join(map(str, B)) + "\n"
    input_str += " ".join(map(str, C)) + "\n"
    
    return input_str

# Generating a test case
X, Y, Z, K, A, B, C = tcgen()
print(format_tc(X, Y, Z, K, A, B, C))
```

**Explanation:**

1. **Random Generation of Parameters:**
    - **\(X\)**, **\(Y\)**, and **\(Z\)** are generated randomly between 1 and 1000.
    - **\(K\)** is generated randomly between 1 and the minimum of 3000 and \(X \times Y \times Z\).
    - Arrays **\(A\)**, **\(B\)**, and **\(C\)** are generated with random values between 1 and \(10^{10}\).

2. **Formatting the Test Case:**
    - The `format_tc` function formats the generated values into the required input format.

3. **Printing the Test Case:**
    - Finally, the test case is generated and printed in the required format.

Run this script to create a new test case each time. This will be useful for testing various scenarios and edge cases for the problem statement.