Problem 57720. The Yellowstone Permutation

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The Yellowstone Permutation is a sequence of positive integers, defined by the following rules:
No term is repeated.
Given n terms, the next term, a(n+1), is always the smallest possible integer.
Every term, a(n), must be relatively prime to the previous term, a(n-1).
Every term, a(n), must have a common divisor greater than 1 with the term before the previous, a(n-2).
The first three terms of the sequence, after which we start applying the rules, are [1  2  3]. 
Given a positive integer, n, return the n-th term of the sequence, a(n).
Example:
n = 4;
a = 4