
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
Given a set \( M \subseteq \{1, 2, \cdots, 2011\} \) that satisfies the property that for any three elements \( a, b, c \) in \( M \), \( a \cdot b \neq c \), we say \( M \) has property \( P \). The set with property \( P \) containing the maximum number of elements is called the maximum set. How many maximum sets with property \( P \) are there? Provide a proof.

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