
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 $g$ be a function taking the positive integers to the positive integers, such that:

(i) $g$ is increasing (i.e. $g(n + 1) > g(n)$ for all positive integers $n$)
(ii) $g(mn) = g(m) g(n)$ for all positive integers $m$ and $n$
(iii) if $m \neq n$ and $m^2 = n^3,$ then $g(m) = n$ or $g(n) = m.$

Find the sum of all possible values of $g(18).$

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