We can write
\begin{align*}
\frac{1}{1 + \omega} + \frac{1}{1 + \omega^2} &= \frac{1 + \omega^2 + 1 + \omega}{(1 + \omega)(1 + \omega^2)} \\
&= \frac{2 + \omega + \omega^2}{1 + \omega + \omega^2 + \omega^3} \\
&= \frac{2 + \omega + \omega^2}{2 + \omega + \omega^2} \\
&= \boxed{1}.
\end{align*}
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $\omega$ be a complex number such that $\omega^3 = 1.$  Find all possible values of
\[\frac{1}{1 + \omega} + \frac{1}{1 + \omega^2}.\]Enter all the possible values, separated by commas.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
1
Let $\omega$ be a complex number such that $\omega^3 = 1.$  Find all possible values of
\[\frac{1}{1 + \omega} + \frac{1}{1 + \omega^2}.\]Enter all the possible values, separated by commas.
test