$($a$)$ We looked at the Three Dining Cryptographers $(3$DC$)$ protocol which can be used to tell whether one of the diners has paid for the meal, but not identify them with certainty. As long as at least two of the cryptographers are honest and do not collude with any other then this is secure$.$ For this question, we add one more party for $4$DC $($A$,$ B$,$ C$,$ and D$)$ in this protocol. The following coin flips are observed by each pair of parties $($Heads$=1,$ Tails$=0)$: AB:$1,$ AC:$1,$ AD:$0,$ BC:$1,$ BD:$0,$ CD:$1.$ The following announcements are made by each party: A:$0,$ B:$0,$ C:$0,$ D:$1.$ The completion of the protocol yields a result of $1.$

i$)$ Consider that we change the protocol. Instead of each party flipping coins with every other party at the table, they now only flip coins with their neighbours on either side of them. Is $4$DC still secure$,$ using the same assumption as before that at least two non$-$colluding cryptographers are present? Explain your reasoning for your answer.