A complex sum!

Algebra Level 5

Let \(\alpha\) be a root of the equation

\[z^{23}=1\]

where \(\alpha\not=1\).

If the sum

\[\displaystyle\sum_{k=0}^{22} \frac{1}{\alpha^{2k}+\alpha^{k}+1}\]

can be expressed as \(\displaystyle\frac{a}{b}\) where \(a\) and \(b\) are co prime positive integers find \(a+b\).

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