Sum of three complex numbers

Question

We have three complex numbers $z_1, z_2, z_3$ such that $|z_1|=|z_3|=|z_3|=1$ and $z_1+z_2+z_3=1$. Find $$S={z_1}^{2013}+{z_2}^{2013}+{z_3}^{2013}$$ I would like only a hint. I probably know everything elementary about complex numbers so just give me a little kick start

Answer 1

In order that both the conditions $|z_1|=|z_3|=|z_3|=1$ and $z_1+z_2+z_3=1$ are satisfied, we require that $z_1,z_2,z_3$ and $1$ form a quadrilateral of four equal sides, therefore a rhombus.

Without loss of generality, we can have $z_1=e^{i\theta}, z_2=1,z_3=-e^{i\theta}$

Can you finish this?