A regular hexagon $ABCDEF$ has side length $18$. Point $P$ lies inside the hexagon so that $AP=12$ and $BP=26$. The centroids of $△PAB$, $△PBC$, $△PCD$, $△PDE$, $△PEF$, and $△PFA$ are the vertices of a convex hexagon. What is the area of that hexagon?