
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 maximal value of positive integer  $n$  such that there exists subset of  $S=\{1,2,...,2001\}$  with  $n$  elements, such that equation  $y=2x$  does not have solutions in set  $S \times S$ 

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