Find m-n

Geometry Level 5

In triangle \(ABC\), \(AC = 13\), \(BC = 14\), and \(AB=15\). Points \(M\) and \(D\) lie on \(AC\) with \(AM=MC\) and \(\angle ABD = \angle DBC\). Points \(N\) and \(E\) lie on \(AB\) with \(AN=NB\) and \(\angle ACE = \angle ECB\). Let \(P\) be the point, other than \(A\), of intersection of the circumcircles of \(\triangle AMN\) and \(\triangle ADE\). Ray \(AP\) meets \(BC\) at \(Q\). The ratio \(\frac{BQ}{CQ}\) can be written in the form \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Find \(m-n\).

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