2021 Fall AMC 12B Problem 16

Suppose a, b, 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?

\textbf{(A)}\: 259\qquad\textbf{(B)} \: 438\qquad\textbf{(C)} \: 516\qquad\textbf{(D)} \: 625\qquad\textbf{(E)} \: 687

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.