Quadrilateral \(ABCD\) has \(AB=CD,\) and the acute angles \(B\) and \(C\) satisfy \(\sin B = \frac{4}{9}\) and \(\sin C= \frac{5}{6}.\) Find \[\frac{\text{area of }\color{red}{\triangle ABC}}{\text{area of }\color{blue}{\triangle BDC}}.\]

Give your answer to 3 decimal places.

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