PUMaC 2012 Number Theory A Problem 7

Let a, b, and c be positive integers satisfying

a^{4}+a^{2} b^{2}+b^{4}=9633

2 a^{2}+a^{2} b^{2}+2 b^{2}+c^{5}=3605

What is the sum of all distinct values of a+b+c?