What's $f(4)$ ?

Algebra Level 4

$\large \begin{cases} a+b+c=0 \\ a^{3}+b^{3}+c^{3}=3 \\ a^{5}+b^{5}+c^{5}=10 \end{cases}$

If $a$ , $b$ and $c$ are real numbers satisfying the system of equations above, find $a^4 + b^4 + c^4$ .