Let $\mathrm{A}_1, \mathrm{~A}_2, \mathrm{~A}_{3, \ldots} \ldots \mathrm{~A}_8$, be the vertices of a regular octagon that lie on a circle of radius 2 . Let $P$ be a point on the circle and let $\mathrm{PA}_{\mathrm{i}}$ denote the distance between the points $P$ and $\mathrm{A}_{\mathrm{i}}$ for $\mathrm{i}=1,2 \ldots \ldots, 8$. If P varies over the circle, then the maximum value of the product $\mathrm{PA}_1 . \mathrm{PA}_2 \ldots \ldots . . \mathrm{PA}_8$, is