In parallelogram $EFGH,$ let $M$ be the midpoint of side $\overline{EF},$ and let $N$ be the midpoint of side $\overline{EH}.$ Line segments $\overline{FH}$ and $\overline{GM}$ intersect at $P,$ and line segments $\overline{FH}$ and $\overline{GN}$ intersect at $Q.$ Find $\frac{PQ}{FH}.$

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