
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
Find all permutations  $(a_1, a_2,...,a_{2021})$  of  $(1,2,...,2021)$ , such that for every two positive integers  $m$  and  $n$  with difference bigger than  $20^{21}$ , the following inequality holds: $GCD(m+1, n+a_1)+GCD(m+2, n+a_2)+...+GCD(m+2021, n+a_{2021})<2|m-n|$ . 

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