$1.$ Design a sequential circuit having one input x so that the circuit generates a signal y$=1$ each time the sub$-$sequence $101011$ is detected.

$($The rightmost$-$end bit is the first one to arrive$)$

$2.$ Design a sequential circuit having an input x$,$ and generating an output signal y $=1$ when the following sub$-$sequence is detected by x: A $0$ followed by any number of $1$s$,$ followed by $00,$ followed by any number of $1$s$,$ and finally followed by $0.$