Trapezoid $A B C D$ has $\overline{A B} || \overline{C D}, B C=C D=43$, and $\overline{A D} \perp \overline{B D}$. Let $O$ be the intersection of the diagonals $\overline{A C}$ and $\overline{B D}$, and let $P$ be the midpoint of $\overline{B D}$.
Given that $O P=11$, the length of $A D$ can be written in the form $m \sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. What is $m+n$ ?