$(20$ marks$)$ Let a LFSR be built with characteristic polynomial $f(x)=x^4+x^3+x^2+x+1.$

$($a$)(10$ marks$)$ Draw a diagram of the LFSR$.$

$($b$)(10$ marks$)$ Show the transition diagram for the LFSR$.$ What is the maximal period of its

output sequences?