
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
Determine all permutations $(a_{1}, a_{2}, \ldots, a_{2021})$ of the sequence $(1,2, \ldots, 2021)$ such that for any positive integers $m$ and $n$ satisfying $|m - n| > 20^{21}$, the following inequality holds: \[\sum_{i=1}^{2021} \operatorname{gcd}(m+i, n+a_{i}) < 2|m-n|\].

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