
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 \( f \) be a function taking positive integers to positive integers, such that:

(i) \( f \) is increasing (\( f(n + 1) > f(n) \) 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 \( f(n) = m \)

Find all possible values of \( f(60) \).

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