Geometry Level 4

The altitudes $$BE$$ and $$CF$$ of a triangle $$ABC$$ meet at $$H$$. The extensions of $$FE$$ and $$BC$$ meet at $$U$$. A line through the midpoint $$L$$ of $$BC$$ parallel to the internal bisector of $$\angle{EUB}$$ meets $$CA, AB, HC, HB$$ (extended as necessary) at $$P, Q, X, Y$$, respectively. If the circumradius of $$APQ$$ is $$r_{1}$$ and the circumradius of $$HXY$$ is $$r_{2}$$, find $$\dfrac{r_{1}}{r_{2}}$$.