Newton or Vieta?

Algebra Level 4

Let \(r_1\), \(r_2\), \(r_3\), ..., and \(r_{20}\) be the roots of the equation \(z^{20}-19z+2=0\).

Find

\[r_1^{20}+r_2^{20}+r_3^{20}+\dots+r_{20}^{20}\]

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