Let $ABCDEF$ be a regular hexagon of side length $2$. Let us construct parallels to its sides passing through its vertices and midpoints, which divide the hexagon into $24$ congruent equilateral triangles, whose vertices are called nodes. For each node $X$, we define its trio as the figure formed by three adjacent triangles with vertex $X$, such that their intersection is only $X$ and they are not congruent in pairs. a) Determine the maximum possible area of a trio. b) Show that there exists a node whose trios can cover the entire hexagon, and a node whose trios cannot cover the entire hexagon. c) Determine the total number of triangles associated with the hexagon.