The following is a circuit that utilizes a multiplexer $($MUX$).$

Consider a circuit where two $4-$bit unsigned integers, A$=$a$3$a$2$a$1$a$0$ and B$=$b$3$b$2$b$1$b$0,$ are input into the multiplexer. In this circuit, when c$4=1,$ A$>$B$,$ and in this case, the output Z is A$.$ Conversely, when c$4=0,$ A$<=$B$,$ and in this case, the output Z is B$.$ Also, set c$0=0.$

Now, consider the logic expressions representing the outputs of circuits N$0$ to N$3.$ Specifically, given that circuits N$0$ to N$3$ are identical, express the output ci$+1$ of circuit Ni in terms of ci$,$ ai$,$ and bi in the minimum sum$-$of$-$products form. Provide intermediate steps using Karnaugh maps in your explanation.