Super fun complex number question

Algebra Level 3

Suppose \(z\) is a complex number such that \(z^5 + 1 = 0.\)

If \(z^4 + z^2 + 1 = z^m + z^n\), then find \(\left|m - n\right|\), where \(m\) and \(n\) are the smallest possible positive integers for which the listed equation is true.

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