
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
Given a rational number \( r > 1 \), Alice plays a game. Initially, on the real line, there is a red piece at position 0 and a blue piece at position 1. In each move, Alice selects an integer \( k \), and she denotes the positions of the two pieces as \( x \) and \( y \) in any order she prefers. She then moves the piece positioned at \( y \) to \( y' \) such that \( y' - x = r^k (y - x) \). Alice wins if she can move the red piece to position 1 in at most 2021 moves.

For which values of \( r \) can Alice win?

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