Consider the following problem from Homework $8.$ During any period, a

potential customer arrives at a certain facility with probability $1/2.$ If there are two people in the facility $($including the one being served$)$ the potential customer leaves the facility immediately and never returns. However, if there is one or fewer people, he enters the facility and becomes an actual customer. The manager of the facility has two types of service rates available. If she uses her slow service rate at a cost of $$3$ during a period, a customer will be served and leave the facility with probability $3/5.$ If she uses her fast service rate at a cost of $$9$ during a period, a customer will be served and leave the facility with probability $4/5.$ Note that the probability of more than one customer arriving or more than one customer being served in a period is $0.$A profit of $$50$ is earned when a customer is served. Using value iteration, compute the optimal policy that maximizes the infinite horizon expected discounted profit with discount factor $\backslash$alpha $=0\mathrm{.}9.$