A complex sum!

Algebra Level 5

Let α\alpha be a root of the equation

z23=1z^{23}=1

where α1\alpha\not=1.

If the sum

k=0221α2k+αk+1\displaystyle\sum_{k=0}^{22} \frac{1}{\alpha^{2k}+\alpha^{k}+1}

can be expressed as ab\displaystyle\frac{a}{b} where aa and bb are co prime positive integers find a+ba+b.

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