Moving Point Perpendicular to a Diagonal Plane in a Rhombic Prism

In the prism $ABCD\text{-}A_1B_1C_1D_1$, the edge $AA_1$ is perpendicular to the base plane $ABCD$, and the base $ABCD$ is a rhombus with $AB = 2$ and $\angle BAD = 60^\circ$; also $AA_1 = \dfrac{5}{2}$. Let $P$ be the midpoint of $BC$, let $M$ be a point on the edge $AA_1$, and let $Q$ be a point moving over the quadrilateral $ABCD$ (interior and boundary), with $Q \neq P$, such that $PQ \perp$ plane $AA_1C_1C$. If the cosine of the angle between the skew lines $PQ$ and $B_1M$ is $\dfrac{\sqrt{2}}{4}$, find the range of possible values of $MQ$.