Suppose you have $$16,000$ to invest in three stocks, A$,$ B$,$ and C$.$ Stock A is a low$-$risk stock that has expected returns of $4\%.$ Stock B is a medium$-$risk stock that has expected returns of $5\%.$ Stock $C$ is a high$-$risk stock that has expected returns of $6\%.$ You want to invest at least $$1,000$ in each stock. To balance the risks, you decide to invest no more than $$7,000$ in stock $C$ and to limit the amount invested in $C$ to less than $4$ times the amount invested in stock $A.$ You also decide to invest less than twice as much in stock B as in stock A$.$ How much should you invest in each stock to maximize your expected profit?