HMMT November 2016 H9

The vertices of a regular nonagon are colored such that 1 ) adjacent vertices are different colors and 2) if 3 vertices form an equilateral triangle, they are all different colors. Let $m$ be the minimum number of colors needed for a valid coloring, and $n$ be the total number of colorings using $m$ colors. Determine $m n$. (Assume each vertex is distinguishable.)