Let \(ABC\) be a triangle with \(m(\widehat{A})=90^\circ\), \(|AB|=4\), and \(|AC|=3\). Let \(D\) be the foot of the perpendicular from \(A\) to \([BC]\). If \(P\) is a point on \([BD]\) such that \(5|AP|=13|PD|\), find the length of \(|CP|\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).

Let's think step by step and output the final answer within \boxed{}.
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