algebra precalculus - Telescoping with imaginary numbers

If $\omega^{1997} = 1$ and $\omega \neq 1$, then
$$\frac{1}{1 + \omega} + \frac{1}{1 + \omega^2} + \dots + \frac{1}{1 + \omega^{1997}}$$
can be written in the form $m/n$, where $m$ and $n$ are relatively prime positive integers. Find the remainder when $m + n$ is divided by 1000.

I really can't seem to find the complex number w that satisfies this condition, and I cannot find any patterns/telescoping methods. Can anyone help me or give me some pointers?

Thanks!