Problem of the Day #28 (Challenge)

POTD Challenge December 17, 2023
Author: excelex

A regular 11-gon $A_1 A_2 A_3... A_{11}$ has circumradius $1$ and circumcenter $O$. Point $P$ is constructed such that $\angle OA_1P=90^{\circ}$, $\angle POA_1= 60^{\circ}$, and $\angle A_1PO=30^{\circ}$ . In other words, $\triangle OPA_1$ is a $30-60-90$ triangle with smallest side being $OA_1$. If $N$ is the value of $\prod_{i=1}^{11} A_i P$, find the remainder when $N^2$ is divided by $1000$.