3 variable diophantine system

Number Theory Level 4

Given the positive integers $a \le b \le c \le 100$ satisfy the system of equations below.

$\begin{cases} {a+b=c} \\ {a^4+b^2=c^2} \\ {a^3=b+c} \\ \end{cases}$

Find the sum of all possible values of $a+b+c$ .