You've already got $|z|=1$ correctly.
Note here that we have to have
$$z\not=z^2,z\not=z^3,z\not=z^4,z^2\not=z^3,z^2\not=z^4,z^3\not=z^4$$
giving
$$\theta\not=\frac{2}{3}\pi,\pi,\frac{4}{3}\pi$$
Also, we have to consider the expression "these taken in order".
For $0\lt\theta\le\frac{\pi}{2}$, since we have $0\lt\theta\lt 2\theta\lt 3\theta\lt 4\theta\le 2\pi$, these $\theta$ are sufficient.
For $\frac{\pi}{2}\lt\theta\lt\frac{2}{3}\pi$, since we have $\frac{\pi}{2}\lt\theta\lt 2\theta\lt 3\theta\lt 2\pi\lt 4\theta\lt 4\pi$ and $0\lt 4\theta-2\pi\lt \theta\lt 2\theta\lt 3\theta\lt 2\pi$, these $\theta$ are sufficient.
For $\frac{2}{3}\pi\lt\theta\lt\pi$, since we have $\frac{2}{3}\pi\lt \theta\lt 2\theta\lt 2\pi\lt 3\theta\lt 4\theta\lt 4\pi$ and $0\lt 3\theta-2\pi\lt\theta\lt 4\theta-2\pi\lt 2\theta\lt 2\pi$, these $\theta$ are not sufficient.
For $\pi\lt\theta\lt\frac{4}{3}\pi$, since we have $\pi\lt\theta\lt 2\pi\lt 2\theta\lt 3\theta\lt 4\pi\lt 4\theta\lt 6\pi$ and $0\lt 2\theta-2\pi\lt 4\theta-4\pi\lt \theta\lt 3\theta-2\pi\lt 2\pi$, these $\theta$ are not sufficient.
For $\frac{4}{3}\pi\lt\theta\lt\frac{3}{2}\pi$, since we have $\frac{4}{3}\pi\lt \theta\lt 2\pi\lt 2\theta\lt 4\pi\lt 3\theta\lt 4\theta\lt 6\pi$ and $0\lt 3\theta-4\pi\lt 2\theta-2\pi\lt \theta\lt 4\theta-4\pi\lt 2\pi$, these $\theta$ are sufficient.
For $\frac{3}{2}\pi\le\theta\lt 2\pi$, since we have $\frac{3}{2}\pi\le \theta\lt 2\pi\lt 2\theta\lt 4\pi\lt 3\theta\lt 6\pi\le 4\theta\lt 8\pi$ and $0\le 4\theta-6\pi\lt 3\theta-4\pi\lt 2\theta-2\pi\lt \theta\lt 2\pi$, these $\theta$ are sufficient.
It follows from these that the answer is
$$\color{red}{0\lt\theta\lt\frac{2}{3}\pi\quad\text{or}\quad \frac{4}{3}\pi\lt\theta\lt 2\pi}$$
