Import Exclusives Ltd$.$ received an invoice from Dansk Specialties of Copenhagen with terms $5/20$ R$.$O$.$G$.$ for the items listed below.

$100$ teak trays at $$30\mathrm{.}30$ each

$20$ teak buckets at $$68\mathrm{.}20$ each

$35$ teak salad bowls at $$55\mathrm{.}40$ each

All items are subject to trade discounts of $35\frac{2}{3}\%,8\frac{1}{2}\%,$ and $6\%.$

$($a$)$ If the shipment was received on June $28,$ what is the last day of the discount period?

$($b$)$ What is the amount due if the invoice is paid in full on July $15?$

$($c$)$ If only a partial payment is made on the last day of the discount period, what amount is needed to reduce the outstanding balance to $$2700?$

$($a$)$ The last day for taking the cash discount is

$($Type a whole number.$)$