3. A robot is placed at point $P$ on a number line, except for points $0$ and $1$, and the robot can only move left or right along the number line. Two players, A and B, play the following game: A gives a distance, and B gives a direction, and the robot will move the specified distance in the specified direction. A's goal is to move the robot to point 0 or 1, while B tries to prevent A from achieving this goal. Question: At which point $P$ does A have a winning strategy?
3. If $P \notin[0,1]$, player B just needs to choose a direction to move the robot away from $[0,1]$ to win;
If $P=0$ or 1, then player A wins;
If $P=\frac{m}{2^{n}}\left(m, n \in \mathbf{Z}_{+}, m<2^{n}\right)$, then player A wins.
Assuming $m$ is odd. Player A chooses $\frac{1}{2^{n}}$, then the robot will move to $\frac{m+1}{2^{n}}$ or $\frac{m-1}{2^{n}}$. If these numbers are neither 1 nor 0, then after removing the common factor, the robot will be at $\frac{k}{2^{l}}\left(k<2^{l}\right.$, $l<n)$ position. Player A then chooses $\frac{1}{2^{l}}$, and so on. Eventually, the denominator becomes 1, and $P$ is 0 or 1.

Assuming $x \in(0,1)$ is not of the form $\frac{m}{2^{n}}$, no matter what distance $d$ player A chooses, at least one of $x+d$ or $x-d$ is not of the form $\frac{m}{2^{n}}$. Player B just needs to choose a direction to move the robot to this number, so the robot will never reach 0 or 1.