$(1)$ To simplify the problem, we believe that there is no turnaround time between flights, which means that the next flight will take off immediately after one flight ends $($if the next flight cannot take off immediately, the airport will wait for the next flight from that city$).$ Establish a network consisting of the feasible routes mentioned above $($note: including nodes every half hour from $8$:$00$ AM to $7$:$30$ PM in each city$).$ By establishing and solving a spreadsheet model, the optimal combination of routes can be determined to achieve maximum profit.$(2)$ To achieve economies of scale, Richard is considering leasing more aircraft. The rental price for each aircraft is $$30000$ per day. Using what if analysis: Is it better to use $5,6,$ or $7$ planes than $4$ planes.$(3)$ When considering the turnaround time for passengers boarding and disembarking between flights $(30$ minutes$),$ reconsider problem $(1)($the actual turnaround time for most aircraft is usually longer$).$ Will this change the number of routes?$(4)$ Richard is now considering not having each plane fly empty to another city, but rather transporting a certain amount of cargo. In this way, not only will there be no cost of $$5000,$ but it will also bring a profit of $$5000.$ Use the spreadsheet in $(3)$ to find the optimal route combination to maximize profits. Will this change the number of flights to another city?