Let $A = (2, 0)$, $B = (0, 2)$, $C = (-2, 0)$, and $D = (0, -2)$.  Compute the greatest possible value of the product $PA \cdot PB \cdot PC \cdot PD$, where $P$ is a point on the circle $x^2 + y^2 = 9$.

Let's think step by step and output the final answer within \boxed{}.
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