GymFit, a popular fitness club chain. Currently offers a single membership plan called GymFit premium. This plan is priced at $$99$ per month and has attracted $50,000$ dedicated members, whom the company internally refers to as Fitness Enthusiasts. Through market research, GymFit$$s pricing analyst, Alex, identifies an additional $75,000$ potential members called Casual Gym$-$Goers, who are willing to pay only $$59/$month$/$ Alex proposes introducing a discounted restricted plan $($restricted access during peak hours, no free nutrition consultation, etc.$)$ called GymFit Basic to target Casual Gym$-$Goers. Alex estimates the following willingness to pay for each plan:

$-$ Fitness Enthusiasts: $$79$ for Basic, $$99$ for Premium

$-$ Casual Gym$-$Goers: $$49$ for Basic, $$59$ for Premium

Variable costs per member are estimated at $$20/$month for the Basic plan and $$30/$month for the Premium plan.

$($A$)$ If Alex wants to optimally segment the market between Fitness Enthusiasts and Casual Gym$-$Goers, what prices should she set for the Basic and Premium plans? Calculate the incremental profit that GymFit will generate from introducing the additional tier compared to the existing single$-$price plan. $($B$)$ Alex$$s proposal is met with skepticism by her manager, Chad Wilson. Chad believes that introducing the Basic plan will lead to $25\%$ of Fitness Enthusiasts downgrading from the Premium plan to the Basic plan. Assuming Chad is correct, calculate the overall profit impact of introducing the tiered pricing structure compared to the existing single$-$price plan.