
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
For positive integers $n$, let $g(n)$ return the smallest positive integer $m$ such that $\frac{1}{m}$ has exactly $n$ digits after the decimal point. Find the number of positive integer divisors of $g(2010)$ if $g(n)=3^n$.

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