Let $ABCD$ be a convex cyclic quadrilateral. Suppose $P$ is a point in the plane of the quadrilateral such that the sum of its distances from the vertices of $ABCD$ is the least. If $\{PA, PB, PC, PD\} = \{3, 4, 6, 8\}$, what is the maximum possible area of $ABCD$?

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