HMMT 2025 T6

Complex numbers $\omega_{1}$ , ..., $\omega_{n}$ each have magnitude 1. Let $z$ be a complex number distinct from $\omega_{1}$ , ..., $\omega_{n}$ such that

$$\frac{z + \omega_{1}}{z - \omega_{1}} +\dots +\frac{z + \omega_{n}}{z - \omega_{n}} = 0.$$

Prove that $|z| = 1$