
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
In $\triangle ABC$, \( AB = 2019 \), \( BC = 2020 \), and \( CA = 2021 \). Yannick draws three regular \( n \)-gons in the plane of \( \triangle ABC \) so that each \( n \)-gon shares a side with a distinct side of \( \triangle ABC \) and no two of the \( n \)-gons overlap. What is the maximum possible value of \( n \)?

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