Question 164
Suppose a quadrilateral $ABCD$ is inscribed in a circle of radius $1$, and its diagonals intersect with the angle of $60^\circ$. Let $P$ be the point of intersection of the diagonals. Suppose that it is known that $AP = \frac{1}{3}$ and $CP = \frac{2}{3}$. Determine all possible values that the absolute value of the difference of $BP$ and $DP$ can take. Here, we represent by $XY$ the length of the line segment $XY$.