A causal LTI system has an impulse response $h[n],$ the $z-$transform of which is $H(z)=\frac{1+z^{-1}}{\{1-(\frac{1}{4})z^{-1}\}\{1+(\frac{1}{2})z^{-1}\}}.$

$($a$)$ What is the ROC of $H(z)?$

$($b$)$ is the system stable? Explain.

$($c$)$ Find the $z-$transform $x(z)$ of an input $x[n]$ that will produce the output $y[n]=-(\frac{1}{2})(\frac{1}{4}{)}^{n}u[n]-(\frac{4}{3})(2{)}^{n}u[-n-1].$

$($d$)$ Find the impulse response $h[n]$ of the system.