The shortest distances between an interior [diagonal](https://artofproblemsolving.com/wiki/index.php/Diagonal) of a rectangular [parallelepiped](https://artofproblemsolving.com/wiki/index.php/Parallelepiped), $P$, and the edges it does not meet are $2\sqrt{5}$, $\frac{30}{\sqrt{13}}$, and $\frac{15}{\sqrt{10}}$. Determine the [volume](https://artofproblemsolving.com/wiki/index.php/Volume) of $P$.