Find that length!

Geometry Level 5

There exists a triangle \(ABC\) with \(AB=5\) and \(AC=7\). Let \(D\) be a point on \(BC\) such that \(CD=2BD\). Given that \(\angle A\) is obtuse, the area of \(\triangle ABC = \frac{21\sqrt{11}}{4}\), and the length of \(AD\) can be expressed in the form \(\sqrt{n}\) where \(n\) is an integer, find the value of \(n\).

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