Find all complex numbers $z$ such that $|z| = 1$ and $\Im((z+1)^{2020}) = 0$

Question

I've stumbled upon an interesting problem. The task is to find all complex numbers $z$ such that $$|z| = 1$$ and $$\Im((z+1)^{2020}) = 0$$
So far I found, that it's possible to follow these steps:
$$u = z+1$$
$$\Im(u^{2020}) = 0$$
$$\sin(2020x) = 0$$
$$x = \dfrac{\pi n}{2020}$$
Which basically gives me all complex numbers $u$ that follow $\Im(u^{2020}) = 0$. However, I don't know how to continue from this point.

Answer 1

From $\Im((z+1)^{2020}) = 0$, it follows that $$\operatorname{arg}(z+1) = \frac{2k\pi}{2020} $$ $z=-1$ is one solution. If not, from $|z| = 1$, it holds that $$\operatorname{arg}(z+1) = \frac{1}{2} \operatorname{arg}(z)$$ Combining the two $$\operatorname{arg}(z) = \frac{k\pi}{505}$$