Let $A _ { 1 } , A _ { 2 } , A _ { 3 } , \ldots , A _ { 8 }$ 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 $i = 1,2 , \ldots , 8$. If $P$ varies over the circle, then the maximum value of the product $P A _ { 1 } \cdot P A _ { 2 } \cdots P A _ { 8 }$, is