Hidden Polynomial! (No easy logic)

Algebra Level 1

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

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

Note: I found the problem in my Book. See: "Ayo Raih Medali Emas Olimpiade Matematika" (English: Let's Get A Gold Medal in Mathematics Olympiad) by: Dr. Khow Yao Tung, M. Sc. Ed, MEd.
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