Let $\omega$ be a complex number such that $\omega^5 = 1$ and $\omega \neq 1$. find \[\frac{\omega}{1 + \omega^2} + \frac{\omega^2}{1 + \omega^4} + \frac{\omega^3}{1 + \omega} + \frac{\omega^4}{1 + \omega^3}.\]