
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  $\alpha\geq 1$  be a real number. Define the set  $$ A(\alpha)=\{\lfloor \alpha\rfloor,\lfloor 2\alpha\rfloor, \lfloor 3\alpha\rfloor,\dots\} $$  Suppose that all the positive integers that **does not belong** to the  $A(\alpha)$  are exactly the positive integers that have the same remainder  $r$  in the division by  $2021$  with  $0\leq r<2021$ . Determine all the possible values of  $\alpha$ .

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