2021 IMO Shortlist N4

Alice is given a rational number $r>1$ and a line with two points $B \neq R$, where point $R$ contains a red bead and point $B$ contains a blue bead. Alice plays a solitaire game by performing a sequence of moves. In every move, she chooses a (not necessarily positive) integer $k$, and a bead to move. If that bead is placed at point $X$, and the other bead is placed at $Y$, then Alice moves the chosen bead to point $X^{\prime}$ with $\overrightarrow{Y X^{\prime}}=r^{k} \overrightarrow{Y X}$. Alice's goal is to move the red bead to the point $B$. Find all rational numbers $r>1$ such that Alice can reach her goal in at most 2021 moves.

给 Alice 一个有理数 $r>1$ 和一条有两个点 $B \neq R$ 的直线，其中点 $R$ 包含一个红色珠子，点 $B$ 包含一个蓝色珠子。爱丽丝通过执行一系列动作来玩纸牌游戏。在每次移动中，她都会选择一个(不一定是正数)整数 $k$ 和一个要移动的珠子。如果该珠子放置在点 $X$，而另一个珠子放置在 $Y$，则 Alice 将所选珠子移动到点 $X^{\prime}$，其中 $\overrightarrow{Y X^{\prime}}=r^{k} \overrightarrow{Y X}$。 Alice 的目标是将红珠移动到点 $B$。找到所有有理数 $r>1$，使得 Alice 最多可以在 2021 步中达到她的目标。