Side Side Angle?

Geometry Level 4

Let \(\triangle ABC\) have \(AB=20\) and \(BC=15\). If \[\text{max}(\tan \angle A)=\dfrac{a\sqrt{b}}{c}\] for positive integers \(a,b,c\) with \(a,c\) relatively prime and \(b\) square-free, find \(a+b+c\).

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