HMMT 2005 T9

Let $A_{1} A_{2} \ldots A_{k}$ be a regular $k$-gon inscribed in a circle of radius 1 , and let $P$ be a point lying on or inside the circumcircle. Find the maximum possible value of $\left(P A_{1}\right)\left(P A_{2}\right) \cdots\left(P A_{k}\right)$.