For a convex quadrilateral $ABCD$, call a point in the interior of $ABCD$ balanced, if (1) $P$ is not on $AC,BD$ (2) Let $AP,BP,CP,DP$ intersect the boundaries of $ABCD$ at $A', B', C', D'$, respectively, then $$AP \cdot PA' = BP \cdot PB' = CP \cdot PC' = DP \cdot PD'$$ Find the maximum possible number of balanced points.