Hidden Polynomial! (No easy logic)

Algebra Level 2

{a+b+c=3a2+b2+c2=5a3+b3+c3=7 \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,ca,b,c are complex numbers satisfying the system of equations above, find the value of a4+b4+c4a^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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