Reminds of Newton?

Algebra Level 4

\[\begin{cases} a+b+c=1 \\ a^2+b^2+c^2=2^2 \\ a^3 + b^3+c^3 = 3^2 \end{cases} \]

Given that \(a,b\) and \(c\) are complex numbers satisfying the system of equations above. If the value of \( \dfrac1{a^3} + \dfrac1{b^3} + \dfrac1{c^3} \) is equal to \( \dfrac pq\), where \(p\) and \(q\) are coprime positive integers, find \(p+q\).

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