Let E be the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$. For any three distinct points $P, Q$ and $Q^{\prime}$ on $E$, let $M(P, Q)$ be the mid-point of the line segment joining P and Q , and $\mathrm{M}\left(\mathrm{P}, \mathrm{Q}^{\prime}\right)$ be the mid-point of the line segment joining P and $\mathrm{Q}^{\prime}$. Then the maximum possible value of the distance between $M(P, Q)$ and $M\left(P, Q^{\prime}\right)$, as $P, Q$ and $Q^{\prime}$ vary on $E$, is $\_\_\_\_$ .