Geometry Problem 1610: Area of a Curvilinear Triangle in a Rhombus

Problem Statement

Let \( ABCD \) be a rhombus with side length \( 6 \) and \( \angle BCD = 60^{\circ} \). Let \( \Gamma_1 \) be the circle centered at \( C \) with radius \( CB \), which intersects the diagonal \( AC \) at point \( E \). Let \( \Gamma_2 \) be the circle centered at \( D \) with radius \( DA \).

Determine the area of the curvilinear triangle bounded by the segment \( AE \), the arc \( BE \) of \( \Gamma_1 \), and the arc \( AB \) of \( \Gamma_2 \) that is exterior to triangle \( ABC \).