Let $z$ be a complex number such that $z^{23} = 1$ and $z \neq 1.$  Find
\[\sum_{n = 0}^{22} \frac{1}{1 + z^n + z^{2n}}.\]
Question:
Let $z$ be a complex number such that $z^{23} = 1$ and $z \neq 1.$  Find
\[\sum_{n = 0}^{22} \frac{1}{1 + z^n + z^{2n}}.\]
Answer:
Let's think step by step.

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Let $z$ be a complex number such that $z^{23} = 1$ and $z \neq 1.$  Find
\[\sum_{n = 0}^{22} \frac{1}{1 + z^n + z^{2n}}.\]
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