AMC 12 2014b 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 $m+n$ ?

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