
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
Define a function $f$ from the positive integers to the positive integers satisfying:
(i) $f(n + 1) > f(n)$ for all positive integers $n$
(ii) $f(mn) = f(m)f(n)$ for all positive integers $m$ and $n$
(iii) if $m \neq n$ and $m^n = n^m$, then $f(m) = n$ or $f(n) = m$
Determine the sum of all possible values of $f(42)$.

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