Consider the following discrete$-$time LTI system

x$[$k $+1]=$

$$

$$

$$

$$

$0100$

$0000$

$0010$

$0011$

$$

$$

$$

$$ x$[$k$]+$

$$

$$

$$

$$

$0$

$1$

$0$

$1$

$$

$$

$$

$$ u$[$k$]$

y$[$k$]=[0110]$ x$[$k$].$

$($a$)$ Classify the four modes of the system into the four categories CO$,$ C $$O$,$CO$,$C $$O$.$

$($b$)$ Is the system $($internally$)$ stable, marginally stable, or unstable?

$($c$)$ Is the system BIBO stable? BIBS stable? state$-$to$-$output stable?

$($d$)$ Assume u$[$k$]0.$ Is it true that y$[$k$]0$ as k $$ for all x$[0]?$

$($e$)$ Assume u$[$k$]0.$ Suppose it is measured that the output y$[$k$]0$ for k $0.$ Can we conclude

that x$[$k$]0?$ If yes, state your reason; if not, find all the possible values of x$[0].$