Let the maximum area of the triangle that can be inscribed in the ellipse \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{4} = 1 \), \( a > 2 \), having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel to the \( y \)-axis, be \( 6\sqrt{3} \). Then the eccentricity of the ellipse is:
