Design a combinational circuit that has $4$ inputs $M(A,B,C,D)$ and $4$ outputs $F(W,x,Y,Z)$ variables. The output of the circuit is defined as follows:

If the MSB and LSB of M are equal, then the F represents the equivalent $2-4-2-1$ code of M$.$

If the MSB and LSB of M are NOT equal, then the F represents the equivalent Excess$-3$ code of M$.$

If the value of $M$ is larger than $9$ then the output $F$ is a don't care.

a$.$ Provide the truth table for the combinational circuit.

b$.$ Write down the SOP expression for each output.

c$.$ Use k$-$maps to simplify the output expressions. Then, if possible, express the outputs obtained from the $k-$maps using $2-$input XOR gates and the minimum number of any additional other gates.

d$.$ Draw the logic diagram for each output