Do You Prefer Quadratics To Cubics?

Algebra Level 2

$\large { \begin{cases} a^3 = a + 1 \\ b^3 = b +1 \\ c^3 = c + 1 \end{cases}}$

If $a,b$ and $c$ are distinct numbers satisfying the system of equations above, find $a^3 + b^3 + c^3$ .