Suppose that you have a delivery business which can run from two warehouses in cities A and B$.$ You only have one delivery truck, so each week you need to choose whether to make deliveries in city A or B$.$ For both warehouses you are relying on daily workers, with the teams of workers being different each week, in each city. Thus, in each week i$,$ paying the worker salaries in city A or B will cost you Ai or Bi $.$ If in week i you run deliveries from a different city from week i$-1,$ then you will need to pay, in week i$,$ for the fuel needed to relocate the truck between warehouses. Assume that this fuel cost F is always the same, as the distance between the warehouses is not changing. However, if in week i you run deliveries from the same city as in the previous week i$-1,$ then in week i you will get a discount D$\_$A or D$\_$B $,$ from the local authorities in city where you are located that week. Given a sequence of n weeks, a plan for your business is a sequence of cities $($A or B$)$ that indicates in which city you performed deliveries for each week. The spending cost of the plan is the sum of salary costs paid in the selected warehouses, along with all the fuel costs incurred whenever the truck moved between cities, and subtracting any discounts received during the course of the plan. Your plan can begin in either of the two cities A or B$.$ The goal of the problem is that given a value for the fuel cost F$,$ discounts D A and D B $,$ and sequences of salary costs A$1,...,$ An and B$1,...,$ Bn $,$ to find a plan with minimum spending cost. Develop a dynamic programming algorithm that finds the value and solution for an optimal plan using the steps outlined below. $($a$)[20$ points$]$ Write a recursive formula for computing the optimal value for the spending cost