OBM 1991 Problem 5

$P_0 = (1,0), P_1 = (1,1), P_2 = (0,1), P_3 = (0,0)$ . $P_{n+4}$ is the midpoint of $P_nP_{n+1}$ . $Q_n$ is the quadrilateral $P_{n}P_{n+1}P_{n+2}P_{n+3}$ . $A_n$ is the interior of $Q_n$ .

Find $\cap_{n \geq 0}A_n$ .