2014 AMC 12B Problem 24

Let ABCDE be a pentagon inscribed in a circle such that AB = CD = 3, BC = DE = 10, and AE= 14. The sum of the lengths of all diagonals of ABCDE is equal to \frac{m}{n}, where m and n are relatively prime positive integers. What is m+n ?

\textbf{(A) }129\qquad \textbf{(B) }247\qquad \textbf{(C) }353\qquad \textbf{(D) }391\qquad \textbf{(E) }421\qquad

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.