Circumcircle

Level pending

In acute \(\triangle ABC\), \(\angle BAC= \tan^{-1} \left ( \dfrac{3 \sqrt{5}}{7} \right ) \) and \(AC= \dfrac{2}{3}AB\). A point \(D\) is chosen on \(AC\) extended such that \(AD= AB\). The circumcircle of \(\triangle CDB\) intersects \(AB\) at two points: \(E\) and \(B\). If \(\dfrac{AE}{AB}= \dfrac{a}{b}\) for some coprime positive integers \(a, b\), find \(a+b+3\).

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