Let $a, b$, and $c$ be positive integers satisfying
$a^4 + a^2b^2 + b^4 = 9633$
$2a^2 + a^2b^2 + 2b^2 + c^5 = 3605$.
What is the sum of all distinct values of $a + b + c$?

Let's think step by step and output the final answer within \boxed{}.
user