Sure, I'll write a test case generator for the given problem. This generator will create random values for \( N \), \( A \), \( B \), \( C \), and the lengths of \( N \) bamboos. Here is the implementation:

```python
import random

def tcgen():
    # Constraints
    N = random.randint(3, 8)
    
    # Ensure 1 ≤ C < B < A ≤ 1000
    C = random.randint(1, 333)
    B = random.randint(C + 1, 666)
    A = random.randint(B + 1, 1000)

    # Generate lengths of bamboos within the range 1 ≤ l_i ≤ 1000
    bamboo_lengths = [random.randint(1, 1000) for _ in range(N)]

    # Output the test case
    print(N, A, B, C)
    for length in bamboo_lengths:
        print(length)

# Example usage
tcgen()
```

### Explanation
- **N**: Randomly chosen between 3 and 8 (inclusive) because \( 3 \leq N \leq 8 \).
- **C**: Randomly chosen between 1 and 333.
- **B**: Randomly chosen between \( C+1 \) and 666 to ensure \( C < B \).
- **A**: Randomly chosen between \( B+1 \) and 1000 to ensure \( B < A \).
- **bamboo_lengths**: List of \( N \) random integers, each between 1 and 1000 (inclusive).

This script ensures that all the constraints are satisfied while generating the test case. You can run the `tcgen()` function to get random test cases for the problem.