$6.$ Answer the following:

a$)$ Use the implication table method to perform state minimization on the following original FSM and produce the reduced FSM state table:

b$)$ Determine the output sequence for the original and reduced machines for the input sequence $100100101011,$ assuming both machines start in state A$.$

If both machines, the original one and the reduced one, are equivalent, then they should both yield the same output sequence for the same input sequence. Thus the purpose of this part is for you to verify $($at least partially"$)$ that the two machines, the original one and the reduced one, are identical in terms of their output performance. $($A complete verification showing equivalency of the two machines may require a more carefully designed, perhaps longer, input sequence; however, you should consider that the input sequence used for verification purposes in this part is of sufficient depth to provide full verification coverage.$)$

Note that the machine under consideration here is a sequence detector; in this particular problem, the machines should both produce an output of $1$ whenever the input sequence $010$ is applied. You should thus keep this in mind as you are generating your output sequences for the two machines, both original and reduced, and use the expected output behavior of this particular sequence detector as a guide to indicate whether you are doing everything correctly or not.

c$)$ What is the minimum number of flip$-$flops needed to implement the original FSM$?$

What is the minimum number of flip$-$flops needed to implement the reduced FSM$?$

Note that both of the minimum flip$-$flop counts you report should be integer$-$valued; e$.$g$.,$ there is no such thing as $4\mathrm{.}3$ flip$-$flops $(!).$

Make it really easy for me to find both of your numerical, integer$-$valued answers to this part.

Also make it really easy for me to distinguish which value corresponds to the original FSM and which value corresponds to the reduced FSM$.$