$(1)$ Design a divide$-$by$-4$ counter circuit by using a$)$ "binary" encoding, and b$)$ "one$-$hot state" encoding.

$(2)$ Design a synchronous sequential circuit that recognizes the occurrence of the pattern $0010$ by observing an input sequence such as $001010101000100101000110010011$ and outputs $1$ when the pattern is found. Use a$)$ Moore FSMs$,$ b$)$ Mealy FSMs$,$ and c$)$ briefly compare the circuits you obtained with each solution.