
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 $P_n = \{1, 2, \ldots, n\}$, where $n \in \mathbb{N}^*$. Let $f(n)$ be the number of sets $A$ that satisfy the following conditions simultaneously:
1. $A \subseteq P_n$;
2. If $x \in A$, then $2x \notin A$;
3. If $x \in \overline{P_n}A$, then $2x \notin \overline{P_n}A$.

(1) Find $f(4)$;
(2) Find the expression for $f(n)$ in terms of $n$.

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