
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let  $r>1$  be a rational number. Alice plays a solitaire game on a number line. Initially there is a red bead at  $0$  and a blue bead at  $1$ . In a move, Alice chooses one of the beads and an integer  $k \in \mathbb{Z}$ . If the chosen bead is at  $x$ , and the other bead is at  $y$ , then the bead at  $x$  is moved to the point  $x'$  satisfying  $x'-y=r^k(x-y)$ .

Find all  $r$  for which Alice can move the red bead to  $1$  in at most  $2021$  moves.

Present the answer in LaTex format: \boxed{Your answer}
user