$(50$p$)$ An assembly line has three serial workstations. For the assembly line to work properly, these workstations must all function properly. To increase the reliability of the assembly line, extra workers may be added to each workstation. It costs $$400$ to add an extra worker to workstation $1,$$$200$ to workstation $2,$ and $$600$ to workstation $3.$ As a function of the number of added workers $($max$.2),$ the reliability of each workstation is given in table below. Use dynamic programming to maximize the probability that the assembly line will work properly, given that $$1200$ is available for extra workers.

a$)(15$p$)$ Clearly indicate stages, states, decisions, and the recursion relation.

b$)(30$p$)$ Calculate the value of $f_1$ either manually or on Excel using formulae $($if solved on Excel, include a screenshot of your final table and submit your Excel file separately$).$

c$)(5$p$)$ State the optimal strategy that maximizes the reliability of the assembly line. $($Multiple strategies may exist.$)$