2021 AMC 12B Problem 24

Let $ABCD$ be a parallelogram with area $15.$ Points $P$ and $Q$ are the projections of $A$ and $C,$ respectively, onto the line $BD;$ and points $R$ and $S$ are the projections of $B$ and $D,$ respectively, onto the line $AC.$ See the figure, which also shows the relative locations of these points. Suppose $PQ=6$ and $RS=8,$ and let $d$ denote the length of $\overline{BD},$ the longer diagonal of $ABCD.$ Then $d^2$ can be written in the form $m+n\sqrt p,$ where $m,$ $n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. What is $m+n+p?$