
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 each positive integer  $n$ , let  $a(n)$  be the number of zeroes in the base 3 representation of  $n$ . For which positive real numbers  $x$  does the series\[
\sum_{n=1}^\infty \frac{x^{a(n)}}{n^3}
\]converge?

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