UMass Airways $($UMA$)$ operates a plane with $140$ seats, servicing the route between Boston and Chicago, all allocated for economy class passengers. The yield management division at UMA seeks support in developing an overbooking strategy

UMA estimates their average ticket revenue to be $$400.$ In the event of denied boarding due to overbooking, UMA compensates customers by refunding the ticket price and offering a complimentary Boston $$ Chicago ticket for the next available flight. Additionally, passengers receive a complimentary entry pass to the luxurious UMA lounge at Logan Airport. The company incurs a cost of $$600$ per customer for the complimentary ticket and lounge access. Remarkably, customers who experience overbooking express high satisfaction with this service, resulting in a "gain$-$of$-$goodwill" valued at $$100$ per customer. Given that no$-$shows are projected to occur uniformly between $4$ and $14,$ how many seats should UMA overbook each day?