Question 94
$\Delta P_1 P_2 P_3$ is an equilateral triangle. For every $n \ge 4$, Xiao Ming may choose $P_n$ to be either the circumcenter or the orthocenter of $\Delta P_{n-3} P_{n-2} P_{n-1}$. Find all positive integers $n$ for which Xiao Ming, by choosing $P_4, \dots, P_n$ suitably, can make $P_n$ the circumcenter of $\Delta P_1 P_2 P_3$.