In the figure, there is a circle of radius $1$ such that the segment $AG$ is diameter and that line $AF$ is perpendicular to line $DC$. There are also two squares $ABDC$ and $DEGF$, where $B$ and $E$ are points on the circle, and the points $A$, $D$ and $E$ are collinear. What is the area of square $DEGF$?