# Problem Description
This appears to be a problem about calculating a special value R based on an input number N. The system shows a pattern of resistors connected in both parallel and series configurations, where N determines the complexity of the circuit. For each value of N, there's a specific formula to calculate the total resistance R.

# Visual Facts
1. The image shows circuit configurations for N=1, N=2, N=3, and a general case N=i
2. Each box contains a resistor value (r) with subscripts
3. For N=1:
   - Has 2 resistors in series, both with value 1
   - R = 1/(1/1) + 1
4. For N=2:
   - Has 2 parallel resistors (r₁=1, r₂=2) followed by 2 series resistors (r₃=1, r₄=2)
   - R = 1/(1/1 + 1/2) + 1 + 2
5. For N=3:
   - Has 3 parallel resistors (r₁=1, r₂=2, r₃=3) followed by 3 series resistors (r₄=1, r₅=2, r₆=3)
   - R = 1/(1/1 + 1/2 + 1/3) + 1 + 2 + 3
6. For N=i:
   - Shows the general pattern with i parallel resistors followed by i series resistors

# Visual Patterns
1. Circuit Structure Pattern:
   - First part: N resistors in parallel
   - Second part: N resistors in series
   
2. Resistor Value Pattern:
   - In parallel section: values range from 1 to N (r₁=1, r₂=2, ..., rₙ=N)
   - In series section: values also range from 1 to N

3. Formula Pattern:
   - First term: 1 divided by sum of reciprocals (1/1 + 1/2 + ... + 1/N)
   - Plus: sum of series values (1 + 2 + ... + N)


# Solution
def solution(N: int) -> float:
    """
    Calculates the R value for a given number N.

    Parameters:
        N (int): The input number.

    Returns:
        float: The R value, rounded to three decimal places.
    """
    # Calculate the parallel resistance
    parallel_reciprocal = sum(1 / i for i in range(1, N + 1))
    parallel_resistance = 1 / parallel_reciprocal if parallel_reciprocal != 0 else 0

    # Calculate the series resistance
    series_resistance = sum(i for i in range(1, N + 1))

    # Total resistance
    total_resistance = parallel_resistance + series_resistance

    # Return result rounded to three decimal places
    return round(total_resistance, 3)


# Test Script
def test():
    test_cases = [
        (1, 2.0),
        (2, 3.667),
        (3, 6.545),
        (4, 10.48),
        (5, 15.438),
        (6, 21.408),
        (7, 28.386),
        (8, 36.368),
        (9, 45.353),
        (10, 55.341),
    ]

    for N, expected_output in test_cases:
        try:
            output = solution(N)
        except Exception as e:
            error_message = (
                f"An exception occurred while running the test case:\n"
                f"`N`: {N}\n\n"
                f"Exception: '{str(e)}'"
            )
            assert False, error_message

        if output != expected_output:
            error_message = (
                f"The following Test Case failed:\n"
                f"`N`: {N}\n\n"
                f"Actual Output: {output}\n"
                f"Expected Output: {expected_output}"
            )
            assert False, error_message

test()