
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
A finite set  $\mathcal{S}$  of positive integers is called cardinal if  $\mathcal{S}$  contains the integer  $|\mathcal{S}|$  where  $|\mathcal{S}|$   denotes the number of distinct elements in  $\mathcal{S}$ . Let  $f$  be a function from the set of positive integers to itself such that for any cardinal set  $\mathcal{S}$ , the set  $f(\mathcal{S})$  is also cardinal. Here  $f(\mathcal{S})$  denotes the set of all integers that can be expressed as  $f(a)$  where  $a \in \mathcal{S}$ . Find all possible values of  $f(2024)$  $\quad$ Proposed by Sutanay Bhattacharya

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