SMO 2019 Problem 3

A robot is placed at point $P$ on the $x$ -axis but different from $(0,0)$ and $(1,0)$ and can only move along the axis either to the left or to the right. Two players play the following game. Player $A$ gives a distance and $B$ gives a direction and the robot will move the indicated distance along the indicated direction. Player $A$ aims to move the robot to either $(0,0)$ or $(1,0)$ . Player $B$ 's aim is to stop $A$ from achieving his aim. For which $P$ can $A$ win?