Optimization & Lagrange Multipliers · Superchallenge problem

Let $A = (1,0)$, $B = (0,1)$, $C = (-1,0)$, $D = (0,-1)$. Find the maximum of $PA\cdot PB\cdot PC\cdot PD$ over all points $P$ of the closed disk $x^2 + y^2 \le 4$.