Trisectors!

Geometry Level 5

Let \(ABCD\) be a square. Points \(E, F, G, H\) are on the line segments \(AB, BC, CD, DA\) respectively such that \(2AE = EB\), \(2BF = FC\), \(2CG = GD\), and \(2DH = HA\). The area bounded by the line segments \(AG, BH, CE, DF\) is a square with side length 1. If \(AB\) can be written as \(a \sqrt{b}\), where \(a\) and \(b\) are positive integers such that \(b\) is square-free, find \(a+b\).