Let ABC be a triangle. Let S be the circle through B tangent to CA at A and let T be the circle through C tangent to AB at A. The circles S and T intersect at A and D. Let E be the point where the line AD meets the circumcircle of \(\Delta ABC\).

Find the ratio \(\frac{AD}{AE}\).

If your answer is of the form \(\frac{ \sqrt a}{b}\), where \(a\) and \(b\) are coprime integers, insert \(a + b\) as your answer.

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