TopsOJ Summer 2025 - Guts Round - Problem 20

If

• x^2 = 2a^2 + 2b^2 - c^2
• y^2 = 2a^2 + 2c^2 - b^2
• z^2 = 2b^2 + 2c^2 - a^2
• (a + b + c)(a + b - c)(a - b + c)(-a + b + c) = 448

then compute the sum of all integer values of (x + y + z)(x + y - z)(x - y + z)(-x + y + z).