Give a complete problem formulation for each of the following $($initial state, goal test, actions, path cost$).$ Choose a formulation that is precise enough to be implemented.

a$.$ Using only four colors, you have to color a planar map in such a way that no two adjacent regions have the same color.

b$.$ You start with the sequence ABABAECCEC, or in general any sequence made from A$,$ B$,$ C$,$ and E$.$ You can transform this sequence using the following equalities: AC $=$ E$,$ AB $=$ BC$,$ BB $=$ E$,$ and Ex $=$ x for any x$.$ For example, ABBC can be transformed into AEC $($using BB $=$ E$),$ and then AC $($using Ex $=$ x$),$ and then E $($using AC $=$ E$).$ Your goal is to produce the sequence E$.$

c$.$ A container ship is in port, loaded high with containers. There $13$ rows of containers, each $13$ containers wide and $5$ containers tall. You control a crane that can move to any location above the ship, pick up the container under it$,$ and move it onto the dock. You want the ship unloaded.