Let $O$ be a fixed point on a plane. $P_1$, $P_2$, $...$ , $P_{2022}$ are $2022$ variable points on the same plane which do not coincide with $O$. Initially, $P_1$ is an arbitrary point and the line segment $OP_{n+1}$ is $\frac{\pi}{1011}$ anticlockwise to the line segment $OP_n$ for $n = 1$, $2$,$...$, $2021$. In other words, $\angle P_nOP_{n+1} = \frac{\pi}{1011}$ in directed angles. A group of points is said to be "perfect" if $O$ is the point such that the sum of the distances from that point to all the points in the group is the smallest possible. For example, the group of points formed by $P_1$, $P_2$, $P_3$, $P_4$, $P_5$ is said to be "perfect" if $$OP_1 + OP_2 + OP_3 + OP_4 + OP_5 \le XP_1 + XP_2 + XP_3 + XP_4 + XP_5$$for any point $X$ on the plane. (a) Prove that the group of points formed by $P_1$, $P_2$, $...$, $P_{2022}$ is "perfect" initially. (b) You are given a task to remove the points $P_1$, $P_2$, $...$, $P_{2022}$ according the following rules. In each step, you could remove one point, and then rotate another point anticlockwise for $\frac{\pi}{3}$ around point $O$. After each step, the group formed by the remaining points on the plane must remain to be "perfect". Devise an algorithm to remove $1348$ points from the initial $2022$ points and prove that it works.