$1.$ For each problem listed below, give a complete problem defining the following.

$1.$ State representation $2.$ Initial state $3.$ Goal state $4.$ Actions $5.$ Path cost

a$.$ 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, and then AC$,$ and then E$.$ your goal is to produce the sequence E$.$

b$.$You have to color the various regions of a planar map $($for example, the map of

the $48$ contiguous states of the United States$)$ using only four colors $($Red$,$ Blue,

Yellow, and Green$),$ in such a way that no two adjacent regions have the same

color.$$