A jewelry firm buys semiprecious stones to make bracelets and rings. The supplier quotes a price of $$8\mathrm{.}20$ per stone for quantities of $600$ stones or more, $$8\mathrm{.}60$ per stone for orders of $400$ to $599$ stones, and $$9\mathrm{.}10$ per stone for lesser quantities. The jewelry firm operates $170$ days per year. Usage rate is $25$ stones per day, and ordering costs are $$39.$

a$.$ If carrying costs are $$2$ per year for each stone, find the order quantity that will minimize total annual cost.

b$.$ If annual carrying costs are $21$ percent of unit cost, what is the optimal order size?

c$.$ If lead time is $3$ working days, at what point should the company reorder?