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)