Length ratios

Level pending

In \(\triangle ABC\), \(AB=2\), \(AC=4\), and \(BC=5\). Let \(B'\) be the reflection of point \(B\) about \(C\), and let \(G\) be the centroid of the triangle. There exist points \(P\) and \(Q\) on \(AG\) and \(B'C\) respectively such that \(\dfrac{AP}{PG}=\dfrac{B'Q}{QC}=3.\) If \(B'P\) and \(AQ\) intersect at \(X\), and \(M\) is the midpoint of \(BC\), \(MX\) can be written in the form \(\dfrac{\sqrt{m}}{n}\) where \(m\) and \(n\) are relatively prime positive integers. Find \(m+n\).

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