Let ${\mathrm{A}}_1,{\mathrm{A}}_2,{\mathrm{A}}_3,\dots ,{\mathrm{A}}_8$ 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 P A_i denote the distance between the points P and A_i for $\mathrm{i}=1,2,\dots ,8$. If P varies over the circle, then the maximum value of the product ${\mathrm{PA}}_1·{\mathrm{PA}}_2\dots .{\mathrm{PA}}_8$, is