Region of convergence of Z-Transform connected area?

Question

Shouldn't the Region of Convergence of the Z transform be a connected area ? In Oppenheim solution manual, I've found this answer of a question that asks to determine the different forms of the impulse response of a certain system

in the (d) option it shows that the ROC might be $\frac{1}{2} < |z| or |z| > 2$ Why is that ?

Answer 1

It looks definitely wrong to me (besides, the last term should be $u[-n-1]$ instead of $u[n-1]$). In any case, the $\frac{2}{3}2^n u[n]$ part only is valid for $|z|>2$, it's makes no sense to state it as valid solution for $|z|>2$ or $|z|<1/2$. Is that from the "official" solution book?