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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