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 $$a^2+b^2+c^2.$$

(第二十四届AIME1 2006 第10题)