IR-MO-R2 2021 Problem 1

There are two distinct Points $A$ and $B$ on a line. We color a point $P$ on segment $AB$ , distinct from $A,B$ and midpoint of segment $AB$ to red. In each move , we can reflect one of the red point wrt $A$ or $B$ and color the midpoint of the resulting point and the point we reflected from ( which is one of $A$ or $B$ ) to red. For example , if we choose $P$ and the reflection of $P$ wrt to $A$ is $P'$ , then midpoint of $AP'$ would be red. Is it possible to make the midpoint of $AB$ red after a finite number of moves?