Problem $1)(25$ Points$)$

Consider an LTI DT system with the following transfer function:

$H(z)=\frac{z^3}{(z+0\mathrm{.}25)(z-1\mathrm{.}5)(z+2)}$

a$)$ Determine, and sketch, all possible regions of convergence for this z$-$transform. For each ROC, determine whether the corresponding system is $($i$)$ causal$/$anti$-$causal$/$non$-$causal and $($ii$)$ stable$/$unstable$.$

b$)$ For the ROC that corresponds to a stable system, determine the corresponding impulse response.