Let $\omega$ be a complex number such that $|\omega| = 1,$ and the equation
\[z^2 + z + \omega = 0\]has a pure imaginary root $z.$  Find $\omega + \overline{\omega}.$
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $\omega$ be a complex number such that $|\omega| = 1,$ and the equation
\[z^2 + z + \omega = 0\]has a pure imaginary root $z.$  Find $\omega + \overline{\omega}.$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
\sqrt{5} - 1
Let $\omega$ be a complex number such that $|\omega| = 1,$ and the equation
\[z^2 + z + \omega = 0\]has a pure imaginary root $z.$  Find $\omega + \overline{\omega}.$
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