Design a synchronous sequential circuit to recognize $1001$ subsequences including overlaps in an input sequence x $=001001001000100100,$ which gives output y $=000001001000000100,$ Design the circuit using JK flip flop and basic gates.

$($a$)$ Draw the state Diagram by assigning state using gray code

$($b$)$ Draw the state table with output and JK flip flop inputs

$($c$)$ Minimize the functions of output and flip flop inputs

$($d$)$ Draw the circuit diagram using block Diagram of JK flip flops and basic gates