CGMO 2022 Problem 7

Let $n \geqslant 3$ be integer. Given convex $n-$ polygon $\mathcal{P}$ . A $3-$ coloring of the vertices of $\mathcal{P}$ is called nice such that every interior point of $\mathcal{P}$ is inside or on the bound of a triangle formed by polygon vertices with pairwise distinct colors. Determine the number of different nice colorings. (Two colorings are different as long as they differ at some vertices.)