MathNet 1991 — Part 01 Question 6

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