The diameter \(AB\) of a circle of radius \(2\) is extended to a point \(D\) outside the circle so that \(BD=3\). Point \(E\) is chosen so that \(ED=5\) and line \(ED\) is perpendicular to line \(AD\). Segment \(AE\) intersects the circle at a point \(C\) between \(A\) and \(E\). Find the area of \(\triangle ABC\). The original answer is in \(\frac{m}{n}\) format, please give the value of \(m + n\).

Let's think step by step and output the final answer within \boxed{}.
user