
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 \( F \) be the set consisting of all functions \( f \) such that \( f: P(S) \rightarrow \mathbb{R} \) and for all \( X, Y \subseteq P(S) \), \( f(X \cap Y) = \min (f(X), f(Y)) \), where \( S \) is a finite set and \( P(S) \) is the power set of \( S \). Find \( \max _{f \in F} \mid \operatorname{Im}(f) \mid \), where \( \operatorname{Im}(f) \) denotes the image set of \( f \).

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