2008 AIME I Problem 10

Let A B C D be an isosceles trapezoid with \overline{A D} \| \overline{B C} whose angle at the longer base \overline{A D} 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{A D}. The distance E F 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.