3 variable diophantine system

Given the positive integers abc100a \le b \le c \le 100 satisfy the system of equations below.

{a+b=ca4+b2=c2a3=b+c \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+ca+b+c.

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