IGO 2020 E5

We say two vertices of a simple polygon are visible from each other if either they are adjacent, or the segment joining them is completely inside the polygon (except two endpoints that lie on the boundary). Find all positive integers $n$ such that there exists a simple polygon with $n$ vertices in which every vertex is visible from exactly $4$ other vertices. (A simple polygon is a polygon without hole that does not intersect itself.) Proposed by Morteza Saghafian