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 $a+b$ .