
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
Define \( f(n) \) as the number of zeros in the base 3 representation of the positive integer \( n \). For which positive real \( x \) does the series 
\[ F(x) = \sum_{n=1}^{\infty} \frac{x^{f(n)}}{n^3} \]
converge?

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