Let $A BC D$ be a parallelogram with area 30. Points $P$ and $Q$ are the projections of $A$ and $C$ , respectively, onto the line $B D$ ; and points $R$ and $S$ are the projections of $B$ and $D$ , respectively, onto the line $A C$ . See the figure, which also shows the relative locations of these points.

Suppose $PQ = 10$ and $RS = 12$ , and let $d^{'}$ denote the length of $\overline{A C}$ , the shorter diagonal of $A BC D$ . Then $(d^{'})^{2}$ can be written in the form $m + n p$ , where $m$ , $n$ , and $p$ are positive integers and $p$ is not divisible by the square of any prime. What is $m + n + p$ ?