Sure, I can help you write a test case generator for the Yamanote-line Game problem. Below is the Python code that generates random test cases based on the problem constraints:

```python
import random

def tcgen():
    # Randomly generate N within the constraint 3 ≤ N ≤ 1,000
    N = random.randint(3, 1000)
    
    # Randomly generate d within the constraint 1 ≤ d ≤ 1,000
    d = random.randint(1, 1000)
    
    # Generate the list of rewards p of size N where each p_i is within the constraint 1 ≤ p_i ≤ 1,000
    p = [random.randint(1, 1000) for _ in range(N)]
    
    # Return the generated test case
    return N, d, p

# Function to format the output in the required way
def format_test_case(N, d, p):
    # First line: N and d separated by a space
    first_line = f"{N} {d}"
    
    # Second line: list of rewards separated by spaces
    second_line = " ".join(map(str, p))
    
    # Combine both lines into the final formatted string
    return f"{first_line}\n{second_line}"

# Example usage
for _ in range(5):  # Generate 5 test cases for demonstration
    N, d, p = tcgen()
    print(format_test_case(N, d, p))
    print()  # Print a newline for better separation between test cases
```

### Explanation
1. **tcgen Function:**
   - Generates a random integer `N` within the range [3, 1000].
   - Generates a random integer `d` within the range [1, 1000].
   - Generates a list `p` of size `N` where each element is a random integer within the range [1, 1000].

2. **format_test_case Function:**
   - Formats the generated test case into the required input format with `N` and `d` on the first line, and the list of rewards `p` on the second line.

3. **Example usage:**
   - Demonstrates generating and printing 5 test cases.

This code will help generate diverse test cases for testing the Yamanote-line Game problem effectively.