AutoCAD Visual LISP / AutoLISP :: Create Sub Areas From Intersecting Polylines

Essentially, I'll be creating a sort of Venn diagram.  So for the purposes of this exercise, let's assume it's the classic Venn diagram of three intersecting circles.  But I need this to work with closed polylines, that may also have arc segments in them.

Anyway, so I have these intersecting areas, and I need to create bounding polylines for each area, and assess the number of overlaps.  So, as in a Venn diagram, the areas where only one circle is present can then get shaded one color, areas where two circles are overlapping can get a different color, and the area where all three circles overlap can get a third color.

I'm clearly going to have to be converting the various boundaries to Regions, and performing some Unions.  It seems I could grab all objects, convert them to Regions 2 at a time, and compare their Union'ed area to their individual area, to determine if they overlapped at all.  With only the 3 cirlces, that would work, but when I have 35 individual polyline areas with arcs, suddenly this method feels like it will break down.