A manufacturer produces three products, A$,$ B$,$ and C$.$ The profits for each unit of $A,B,$ and $C$ sold are $$1,$$$2,$ and $$3,$ respectively. Fixed costs are $$17,000$ per year, and the costs of producing each unit of $A,B,$ and $C$ are $$4,$$$5,$ and $$7,$ respectively. Next year, a total of $10,000$ units of all three products is to be produced and sold, and a total profit of $$19,000$ is to be realized. If total cost is to be $$68,000,$ how many units of each of the products should be produced next year?

units of product $A,$ units of product $B,$ and $$ units of product $C$ should be produced.