3 variable diophantine system

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 \(a+b+c\).

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