CN-TST 2023 Problem 15

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.