Let \( ABCD \) and \( AEFG \) be squares of side \( 4 \) and \( 2 \) units, respectively. The point \( E \) is on the line segment \( AB \) and the point \( F \) is on the diagonal \( AC \). Then the radius \( r \) of the circle passing through the point \( F \) and touching the line segments \( BC \) and \( CD \) satisfies:
