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