Apple has just introduced iPhone $16$ Pro. You have two options to buy the phone from Apple. The first option

is the upgrade program, which allows you to upgrade to a new unlocked iPhone Pro every year with a monthly

payment of $$60.$ This option comes with an AppleCare$+($a damage replacement plan$).$ The second option

requires you to pay $$1199$ in front with a resale value of $$650$ after one year and $$500$ after two years. In this

case, you need to pay $$199$ extra for the AppleCare$+$ each year. If you do not care whether you can have a new

phone every year or every two years, you can have two choices to own the iPhone. Choice I: you can use the

upgrade program; Choice II: you can pay in full in front, purchase the AppleCare$+$ each year, and in two years,

you will sell your old phone and buy a new one. Assume that you will continue with the same choice going

forward. The appropriate discount rate is $6\%$ APR on a monthly basis.

$($a$)$ What is the present value of the total costs for each choice? Can you compare the PVs to make a decision,

and why?