Let $\omega$ be a nonreal root of $z^3 = 1.$  Find the number of different possible values of
\[(\omega + 1)^n,\]where $n$ is a positive integer.
Question:
Let $\omega$ be a nonreal root of $z^3 = 1.$  Find the number of different possible values of
\[(\omega + 1)^n,\]where $n$ is a positive integer.
Answer:
Let's think step by step.

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6
Let $\omega$ be a nonreal root of $z^3 = 1.$  Find the number of different possible values of
\[(\omega + 1)^n,\]where $n$ is a positive integer.
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