Hidden Polynomial! (No easy logic)

Algebra Level 2

$\large \begin{cases}{a+b+c=3} \\ {a^2 +b^2 +c^2 =5} \\ {a^3 + b^3 +c^3 = 7} \end{cases}$

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