A sequential circuit has one input $(x)$ and one output $(Y).x$ represents a $4-$bit binary number $N,$ which is input least significant bit first. The circuit sets $Y=1$ when it detects the pattern $\text{'}1101\text{'}$ in input $x($rightmost digit is the least significant bit$).$

$($a$)$ Derive a Moore state graph and table with a minimum number of states.

$($b$)$ Try to choose a good state assignment. Explain why you have used that assignment.