Are they all close to power s of 2?

Algebra Level 3

Let \(a,b\) and \(c\) be roots of \(P(x) = x^3 + 3x^2 + 4x - 8\). Suppose \(S_n= a^n + b^n + c^n \). If \(S_3 = 2^m + 1\) and \(S_5 = 2^n + 1 \), find the value of \(m+n\).

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