Eight circles of diameter 1 are packed in the first quadrant of the coordinate plane as shown. Let region $ \mathcal{R} $ be the union of the eight circular regions. Line $ l, $ with slope 3, divides $ \mathcal{R} $ into two regions of equal area. Line $ l $'s equation can be expressed in the form $ ax=by+c, $ where $ a, b, $ and $ c $ are positive integers whose greatest common divisor is 1. Find $ a^2+b^2+c^2. $
[Asymptote diagram]