A Strange Cevian

Geometry Level 2

Triangle \(ABC\) is such that \(AB = 13, BC = 14,\) and \(CA = 15.\) A point \(D\) on \(BC\) is placed such that \(AB + BD = AC + CD.\) Let \(X\) be the intersection of \(AD\) with the incircle of \(ABC\) closest to \(A.\) Find the length of \(BX.\)

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