Let $z \in \mathbb{C}$ be a solution of

$$z^{2}+b z+1=0$$

where $b \in \mathbb{R}$ and $|b| \leqslant 2$. For which values of $b$ do the following hold?

(i) $\left|e^{z}\right|<1$.

(ii) $\left|e^{i z}\right|=1$.

(iii) $\operatorname{Im}(\cosh z)=0$.