Let \(ABC\) be a triangle with \(BC = 20\) and \(CA = 16\), and let \(I\) be its incenter. If the altitude from \(A\) to \(BC\), the perpendicular bisector of \(AC\), and the line through I perpendicular to \(AB\) intersect at a common point, then the length \(AB\) can be written as \(m+\sqrt{n}\) for positive integers \(m,n\). What is \(100m+n?\)

Try my set JEE ADVANCED 2016

This problem is from the Spring OMO 2016

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