In a circle centered at \( O \), \( AB \) is a diameter, and \( C \) is a point on the circle such that \( \angle ACB = 90^\circ \). \( D \) is the midpoint of \( AC \), and \( E \) is the midpoint of \( BC \). If the angle bisector of \( \angle AOB \) intersects \( DE \) at point \( F \), what is the measure of \( \angle AOF \), in degrees? Let's think step by step and output the final answer within \boxed{}.
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