Let \(ABCD\) be a square with side length \(1\). Let \(E\) be a point on side \(BC\) such that \(BE = EC\). Similarly, let \(F\) and \(G\) be points on side \(CD\) such that \(DF = FG = GC\). Let the intersection of \(AF\) and \(DE\) be \(H\) and the intersection of \(AG\) and \(DE\) be \(I\).
Define \(G_1\) to be the centroid of \(\triangle ADH\) and \(G_2\) to be the centroid of \(\triangle AHI.\) If \(G_1G_2\) can be written in form \(\frac{\sqrt{a}}{b}\) where \(a\) is a positive integer not divisible by the square of any prime and \(b\) is a positive integer, find \(a + b\).