Question 10
Let $ABCD$ be an isosceles trapezoid with $\overline{AD} \parallel \overline{BC}$ whose angle at the longer base $\overline{AD}$ is $\frac{\pi}{3}.$ The diagonals have length $10\sqrt{21},$ and point $E$ is at distances $10\sqrt{7}$ and $30\sqrt{7}$ from vertices $A$ and $D,$ respectively. Let $F$ be the foot of the altitude from $C$ to $\overline{AD}.$ The distance $EF$ can be expressed in the form $m\sqrt{n},$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m + n.$