$(28\%)$ The maintenance department of a large hospital uses about $816$ cases of liquid cleanser annually. Ordering costs are $$12,$ carrying costs are $$4$ per case a year, and the new price schedule indicates that orders of less than $50$ cases will cost $$20$ per case, $50$ to $79$ cases will cost $$18$ per case, $80$ to $99$ cases will cost $$17$ per case, and larger orders will cost $$16$ per case. Determine the optimal order quantity and the total cost.