Strange quadrilateral

Geometry Level 3

\(ABCD\) is a square with side length 1. \(E\) divides side \(AD\) into two equal parts. Circle \(k\) centered at point \(F\) is tangential to sides \(BC\), \(CD\) and segment \(BE\).

If \(G\) is the point where circle \(k\) touches \(BE\), find the area of quadrilateral \(EGFD\).

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