Look at the letters used.
There are seven unique letters, AEIMNRS, which could form three possible words as anagrams: REMAINS, MARINES, SEMINAR.
Look at the pattern drawn on the back.
It's a seven-pointed star formed by joining each point to the one two along from it. (Since seven is prime, there are different ways of forming a seven-pointed star, by joining each point to the one two or three along from it. Five is equivalent to two, four to three, and one and six simply give heptagons.)
There's a connection given by
the number seven: seven letters, seven-pointed star. The only question is how to order the letters.
Maybe the three-letter strings are given by
putting the letters in one of the three possible word orders and going along taking every (second, third, nth) letter.
RNA from REMAINSREMAINS (every 4th letter)
IRE from REMAINSREMAINS (every 3rd letter)
RAM from MARINESMARINESMARINES (every 6th letter)
SIM from REMAINSREMAINSREMAINS (every 5th letter)
NES from MARINES (every letter)
NAE from REMAINSREMAINSREMAINS (every 5th letter)
IMN from MARINESMARINES (every 4th letter)
RAS from REMAINS (every 3rd letter)
RAN from SEMINARSEMINARSEMINAR (every 6th letter)
AME from REMAINSREMAINSREMAINS (every 6th letter)
ASM from REMAINSREMAINS (every 3rd letter)
REM from REMAINS (every letter)
The only one not expressible in this form is
SIN. Maybe this is the solution, as there are seven deadly sins?