How do I solve this complex numbers question?

Question

In the solution to a question in the Harvard MIT tournament, it is given that $$\sum_{k=1}^{1006} |z^{2k+1}-z^{2k-1}| = \sum_{k=1}^{1006} |z^{2k}-z^{2k-2}|$$. The next sentence says that this implies that $$|z|\sum_{k=1}^{1006} |z^{2k}-z^{2k-2}|= \sum_{k=1}^{1006} |z^{2k+1}-z^{2k-1}| $$ which implies |z|=1. Why is this so? How did they get this result? For reference, it is the 4th question in the first section which is the Algebra test in https://www.hmmt.co/static/archive/february/solutions/2012/algebra.pdf

Answer 1

We have $\vert zw\vert=\vert z\vert\cdot\vert w\vert$. Apply it to $w=z^{2k}- z^{2k-2}$.

Answer 2

It would be more accurate to say that the second equation is trivial for any $z\in\mathbb{C}$, but combining it with the first equation gives $(|z|-1)w=0$, where $w$ is a non-negative sum of moduli, including $|z^2-1|$ (which vanishes only if $z=\pm 1$).