Inspired by Euler

Geometry Level 5

Given triangle \(ABC\) with \(AB=20, AC=34\), define \(D\in BC\) such that \(AD\perp BC\) and \(AD=16\). \(P\) is a point on segment \(AD\) such that if \(K\) is the foot of projection from \(P\) to \(AB\), then \[OP^2=R(R-2PK)\]where \(O,R\) are the circumcenter and circumradius respectively.

Find the length of \(AP\).

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