In each vertex of a regular $n$-gon, there is a fortress. At the same moment, each fortress shoots one of the two nearest fortresses and hits it. The result of the shooting is the set of the hit fortresses; we do not distinguish whether a fortress was hit once or twice. Let $P(n)$ be the number of possible results of the shooting. Prove that for every positive integer $k\geqslant 3$, $P(k)$ and $P(k+1)$ are relatively prime. (4th Middle European Mathematical Olympiad, Team Competition, Problem 3)