Certainly! To generate test cases for the given problem, we need to ensure that the constraints and input format are adhered to. Here's a Python function that will generate random test cases for the given problem:

```python
import random

def tcgen():
    # Ensure N is between 3 and 8 inclusive
    N = random.randint(3, 8)
    
    # Generate A, B, C such that 1 <= C < B < A <= 1000
    A = random.randint(3, 1000)
    B = random.randint(2, A - 1)
    C = random.randint(1, B - 1)
    
    # Generate the lengths of the bamboos such that 1 <= l_i <= 1000
    lengths = [random.randint(1, 1000) for _ in range(N)]
    
    # Format the output as specified in the problem statement
    input_str = f"{N} {A} {B} {C}\n" + "\n".join(map(str, lengths))
    return input_str

# Example of generating a test case
for _ in range(3):  # Generate 3 sample test cases
    print(tcgen())
    print()
```

### Explanation:
1. **Ensure \(N\) is between 3 and 8 inclusive:**
   ```python
   N = random.randint(3, 8)
   ```

2. **Generate \(A\), \(B\), \(C\) such that \(1 \leq C < B < A \leq 1000\):**
   - First, generate \(A\) such that \(A \leq 1000\):
     ```python
     A = random.randint(3, 1000)
     ```
   - Then generate \(B\) such that \(B < A\):
     ```python
     B = random.randint(2, A - 1)
     ```
   - Finally, generate \(C\) such that \(C < B\):
     ```python
     C = random.randint(1, B - 1)
     ```

3. **Generate the lengths of the bamboos such that \(1 \leq l_i \leq 1000\):**
   ```python
   lengths = [random.randint(1, 1000) for _ in range(N)]
   ```

4. **Format the output as specified in the problem statement:**
   ```python
   input_str = f"{N} {A} {B} {C}\n" + "\n".join(map(str, lengths))
   ```

This function will generate valid random test cases for the given problem, ensuring that all constraints are satisfied. You can use this to test your solution thoroughly.