Silky Smooth lotions come in three sizes: $4,8,$ and $12$ ounces. The following table summarizes the selling prices and variable costs per case of each lotion size.

Per Case $4$ Ounces $8$ Ounces $12$ Ounces

Price $$36\mathrm{.}00$ $$66\mathrm{.}00$ $$72\mathrm{.}00$

Variable cost $$13\mathrm{.}00$ $$24\mathrm{.}50$ $$27\mathrm{.}00$

Fixed costs are $$771,000.$ Current production and sales are $2,000$ cases of $4-$ounce bottles; $4,000$ cases of $8-$ounce bottles; and $1,000$ cases of $12-$ounce bottles. Silky Smooth typically sells the three lotion sizes in fixed proportions as represented by the preceding sales amounts.

Required:

How many cases of $4-,8-,$ and $12-$ounce lotion bottles must be produced and sold for Silky Smooth to break even, assuming that the three sizes are sold in fixed proportions?