Counting Bisecting-Midpoint Values

In a cube $ABCD\text{-}A_1B_1C_1D_1$, let $P$ be a moving point on the four sides of the square $A_1B_1C_1D_1$, $O$ the center of the square $ABCD$, and $M,N$ the midpoints of $AB,BC$. A point $Q$ in the plane $ABCD$ is such that the segments $D_1Q$ and $OP$ bisect each other. Find the number of values of the real number $\lambda$ for which $\overrightarrow{MQ}=\lambda\overrightarrow{MN}$.