$(15)$ A causal system of input $x[n]$ and output $y[n]$ are related through the block$-$diagram representation shown below. D represents a unit delay and a number on the line represents a gain.

$($a$)(5)$ Determine the difference equation relating $x[n]$ and $y[n].$

$($b$)(5)$ Represent the difference equation in $($a$)$ in z$-$domain using z$-$transform.

$($c$)(5)$ Determine the transfer function $H(z)$ and explain whether the system is stable or not.