Problem $1.$ The Galactic Outpost Problem. You are managing a network of research outposts on two planets: Nova Prime $($NP$)$ and Solaris $($S$).$ Each month, you must decide which outpost will serve as your base of operations to support intergalactic experiments. Here$$s the situation: $$ If you operate from Nova Prime in month i$,$ it will cost Ni energy units. $$ If you operate from Solaris in month i$,$ it will cost Si energy units. $$ Moving your operations between the two planets incurs a fixed teleportation cost of M energy units. Your task is to plan your monthly base locations for n months to minimize the total energy cost. This includes the operational energy costs and the teleportation costs between the two planets. You can start on either Nova Prime or Solaris. Example Scenario: Imagine planning for n $=4$ months with the following data: $$ Teleportation cost: M $=10$ Monthly operational energy costs: Month $1$ Month $2$ Month $3$ Month $4$ Nova Prime $($Ni$)132030$ Solaris $($Si$)502024$ The optimal plan would base operations on Nova Prime for months $1$ and $2,$ then move to Solaris for months $3$ and $4.$ This results in a total energy cost of $1+3+2+4+10=20.$ a$)$ A Flawed Approach: Prove the following algorithm doesn$$t always produce the optimal solution. Create an example where it fails, and explain why its result differs from the correct answer. $1$: for i $=1$ to n do $2$: if Ni $<$ Si then $3$: Choose $$Nova Prime in month i$4$: else $5$: Choose $$Solaris in month i$1$ Discussion $62$ b$)$ Minimum Three Teleportations: Construct an example where any optimal plan requires at least three teleportations. Provide a brief explanation of why this is necessary to minimize energy costs. c$)$ The Optimal Plan: Design an efficient algorithm that calculates the minimum total energy cost, given n$,$ M$,$ and the sequences of operational costs N$1,...,$ Nn and S$1,...,$ Sn$.$