MathNet 2021 — Part 10 Question 5

Suppose $a$, $b$, and $c$ are positive integers such that $a + b + c = 23$ and $\gcd(a, b) + \gcd(b, c) + \gcd(c, a) = 9$. What is the sum of all possible distinct values of $a^2 + b^2 + c^2$?