A potential vulnerability was found in the existing nuclear launch control. You are contracted to design a simple and elegant solution that does not require a computer. There are three people whose opinions are important in this regard and they are the President, the Secretary of Defense and the Secretary of State. The Secretary of Defense and the Secretary of State have an independent up or down vote on a possible attack. The President also has a vote, but it can be $$up$,$down$$ or $$neutral$.$ The new rules for triggering the nuclear attack are as follows. If the president votes $$down$,$ there should not be a nuclear attack independent of what the Secretary of Defense and Secretary of State think. If the president votes $$up$,$ the nuclear attack should be triggered as long as at least one of Secretary of Defense or Secretary of State also agrees $($this is to guard against a president unilaterally ordering a nuclear attack$).$ If the president votes $$neutral$,$ the nuclear attack should be triggered only if both the Secretary of Defense and the Secretary of State cast an $$up$$ vote. You have been hired by the Pentagon to design a $4-$input, $1-$output combinational logic circuit whose output will be used to trigger or avoid the nuclear attack. We will designate the output line as A $($attack$)$ and the four input lines as P$,$ Q$,$ D and S$.$ D represents the up or down vote by the Secretary of Defense $(1$ means $$up$$ and $0$ means $$down$).$ S represents the up or down vote by the Secretary of State $(1$ means $$up$$ and $0$ means $$down$).$ Note that, as the President$$s vote has three possible values, we cannot model it using one bit. So$,$ we shall use two bits to model the President$$s vote. Let us call the two bits P and Q$.$ If they both have values of $0,$ it means the President$$s vote is $$down$,$ if they both have values of $1,$ it means President$$s vote is $$up$$ and if P is $0$ and Q is $1,$ it means the President$$s vote is $$neutral$.$ Note that this means the possibility of P being $1$ and Q being $0$ cannot arise at all, and should be utilized for simplification. $1)$ Design an initial circuit whose four inputs are P$,$ Q$,$ S and D and whose output is A$.$ Use only AND Gates, OR Gates and Inverters in your circuit. Draw the initial $($un$-$simplified$)$ circuit $2)$ Translate the required behavior into a truth table. Show all work. $3)$ Show the un$-$simplified SOP expression based on the truth table. This should be in the SOP form: A $=?+?+?+...4)$ Determine a minimum sum of products $($SOP$)$ boolean expression for the output by simplifying algebraically. Show all work. This should be in the SOP form: A $=?+?+?+...5)$ Draw a Karnaugh map and use to simplify the initial circuit. Both equations $($K$-$map $/$ algebra$)$ do not need to match $$ one might be more simplified more than the other. Show loops, what each loop represents, and all work. $6)$ Design a circuit based on your optimal boolean expression $($based on #$4$ or #$5$ above$).$ test it thoroughly to be sure it works as intended. Note: test by stepping through the truth table.