In the circuit given above, considering that $(4Y3Y2Y1Y)=(4A3A2A1A)$ for $S=0$ and $(4Y3Y2Y1Y)=(4B3B2B1B)$ for $S=1,$ and assuming that $(A4A3A2A1)=1010$ and $(B4B3B2B1)=0100$ :

a$)$ Determine $(Y4Y3Y2Y1)$ and $C4,$ if $S=0.$

b$)$ Determine $(Y4Y3Y2Y1)$ and $C4,$ if $S=1.$

c$)$ What kind of circuit is this?