
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  $n$  be a positive integer. Suppose for any  $\mathcal{S} \subseteq \{1, 2, \cdots, n\}$ ,  $f(\mathcal{S})$  is the set containing all positive integers at most  $n$  that have an odd number of factors in  $\mathcal{S}$ . How many subsets of  $\{1, 2, \cdots, n\}$  can be turned into  $\{1\}$  after finitely many (possibly zero) applications of  $f$ ?

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