An Equal-Volume Point Distance

In a pyramid $P\text{-}ABCD$, $AB=AD=\sqrt{10}$, $CB=CD=5$, $\angle BAD=90^\circ$, $PB=4$, $PC=3$. A point $Q$ inside $\triangle PBC$ is such that the pyramid $Q\text{-}ABCD$ and the tetrahedron $Q\text{-}PAD$ have equal volumes. Find the minimum value of $PQ$.