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 $N_i$ energy units.

$-If$ you operate from Solaris $in$ month $i,it$ will cost $S_i$ energy units.

$-$Moving your operations between the two planets incurs a fixed teleportation cost $ofM$ energy units. Your task $isto$ 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:

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 $of1+3+2+4+10=20.$

$a$