Design a combinational circuit that accepts a $4-$bit number A$3$A$2$A$1$A$0$ and generates a $3-$bit binary number output B$2$B$1$B$0$ that approximately the square root of is $<3\mathrm{.}5$ and $>=2\mathrm{.}5$ give a result of $3.$ A$3$ and B$2$ are the most significant bits, respectively. Please describe the optimization process of the design such as a truth table, K$-$maps and the Boolean equation for each output.