AMO 2021 - Secondary 2 (Grade 8) Question 14

In the figure, $ABC$ is equilateral with $AB=1$ foot. Points $D,E$ are the midpoints of $AC,BC$. Segment $FG$ is half as long as $AB$, and $EF=\frac{\sqrt[4]3}{2}$ feet. Find $DH$. As marked, $F,G$ lie on $AB$ and $DH\perp EF$ at $H$.