We control the inventory of a product over $3$ time periods. The inventory capacity is $3,$ and demand can take values $0,1$ or $2$ with probabilities $0\mathrm{.}25,0\mathrm{.}5$ and $0\mathrm{.}25$ respectively. For each unit sold we generate a revenue of $$8$ and for each unit ordered we incur a cost of $$2.$ Every time we place an order we incur a fixed cost $$4.$ Unit inventory holding cost is $$1,$ and demand that is not satisfied is lost with no penalty cost. If the initial inventory level at the beginning of time period $1$ is $0$ units, what is the optimal total expected profit over the whole planning horizon?