
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
Let \( S = \{1, 2, \cdots, n\} \). Define \( A \) as a collection of some subsets of \( S \) such that no two elements in \( A \) are mutually inclusive as subsets of \( S \). Find the maximum number of elements \( |A| \) can have.

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