
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  $\mathbb N$  denote the set of positive integers, and for a function  $f$ , let  $f^k(n)$  denote the function  $f$  applied  $k$  times. Call a function  $f : \mathbb N \to \mathbb N$  *saturated* if \[ f^{f^{f(n)}(n)}(n) = n \] for every positive integer  $n$ . Find all positive integers  $m$  for which the following holds: every saturated function  $f$  satisfies  $f^{2014}(m) = m$ .

*Proposed by Evan Chen*

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