Question $3$: $($worth $54$ points$)$ An airline is preparing to replace its old planes with three new styles of jets. The airline needs $17$ new planes to service its current routes. The decision regarding which planes to purchase should balance cost with capability factors, including the following: $(1)$ The airline can finance up to $$700$ million in purchases; $(2)$ each $7$A$7$ jet will cost $$38$ million, each $7B7$ jet will cost $$27$ million, and each $C_7$ jet will cost $$22$ million; $(3)$ at least one$-$third of the planes purchased should be the longer$-$range $7A7$;$(4)$ the annual maintenance budget is to be no more than $$12$ million; $(5)$ the annual maintenance cost per $7A7$ is estimated to be $$800,000,$$$600,000$ for each $7B7,$ and $$500,000$ for each $7$C$7$; and $(6)$ annually, each $7$A$7$ can carry $125,000$ passengers, each $7$B$7$ can fly $95,000$ passengers, and each $7$C$7$ can fly $80,000$ passengers.

Solve this IP problem to maximize the annual passenger$-$carrying capability. You may want to formulate for yourself as a first step, but I don't need to see that.