
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  $f(n)=\varphi(n^3)^{-1}$ , where  $\varphi(n)$  denotes the number of positive integers not greater than  $n$  that are relatively prime to  $n$ . Suppose
\[ \frac{f(1)+f(3)+f(5)+\dots}{f(2)+f(4)+f(6)+\dots} = \frac{m}{n} \]
where  $m$  and  $n$  are relatively prime positive integers. Compute  $100m+n$ .

*Proposed by Lewis Chen*

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