a$)$ Using a J$-$K flip$-$flop design a synchronous counter which will detect the input sequence OIOI. The sequence may overlap as shown below: Your solution should provide following: state diagram, state table, derivation of next$-$state and output expressions and the implementation circuit diagram.

$x=010101001101011$

$z=000101000000010$

$[20$marks$]$