Rory Company has an old machine with a book value of $$77,000$ and a remaining five$-$year useful life. Rory is considering purchasing a new machine at a price of $$103,000.$ Rory can sell its old machine now for $$66,000.$ The old machine has variable manufacturing costs of $$33,000$ per year. The new machine will reduce variable manufacturing costs by $$13,200$ per year over its five$-$year useful life.

$($a$)$ Prepare a keep or replace analysis of income effects for the machines.

$($b$)$ Should the old machine be replaced?