Right-Triangle Product

Find positive integers a < b < c with a² + b² = c² and a + b + c = perimeter. Return the largest product a × b × c among qualifying triangles, or −1 if there are none. This generalizes the fixed-perimeter source task and explicitly handles multiple answers.

Implement pythagorean_product(perimeter).

Input

perimeter: The required sum of the three sides.

Output

Return the largest product, or −1 when no triangle qualifies.

Constraints

3 ≤ perimeter ≤ 100,000. Products fit signed 64-bit integers.

Examples

Input:
perimeter = 30

Expected output:
expected = 780

Input:
long long perimeter = 30;

Expected output:
long long expected = 780;

Input:
long perimeter = 30;

Expected output:
long expected = 780;

Input:
perimeter = 11

Expected output:
expected = -1

Input:
long long perimeter = 11;

Expected output:
long long expected = -1;

Input:
long perimeter = 11;

Expected output:
long expected = -1;

References

This challenge is adapted for CodeUtility. Follow the statement and constraints above.

• #9 Special Pythagorean Triplet