P$2(15$ points$).$ Consider the Boolean following Boolean function in the image.

a$.$ Draw the truth table for this function.

b$.$ Use the hierarchical divide$-$and$-$conquer method to partition the truth table. Write the value of the function for each of the smallest partitions. This will help you design the circuit in part c $.$ Hint: the first partition is by A ; the second partition is by B $,$ but it is performed twice.

c$.$ Draw a logic circuit that implements this function using only $2-$to$-1$ multiplexers. No other logic gates are allowed. If you need to negate a signal you must use a multiplexer for that as well. Clearly label all signals and show the connections between the multiplexers.