Problem of the Day #35

POTD January 9, 2024
Note that POTD structure was changed again to have a random point difficulty every day
Author: awesomeming327

Three nonoverlapping regular hexagons share a vertex. Max stands at a random point on the same plane. The expected value of the number of hexagon vertices he can see (if two hexagons have a vertex in common, that vertex counts as one vertex rather than two) can be expressed as \frac{m}{n} where m and n are relatively prime positive integers. Find m + n.