
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 the maximum value of the positive integer \( n \) such that for any simple graph of order \( n \) with vertices \( v_1, v_2, \ldots, v_n \), one can always find \( n \) subsets \( A_1, A_2, \ldots, A_n \) of the set \( \{1, 2, \ldots, 2020\} \) which satisfy: \( A_i \cap A_j \neq \varnothing \) if and only if \( v_i \) and \( v_j \) are adjacent.

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