
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 \( b > 1 \) be an integer. For any positive integer \( n \), let \( d(n) \) be the number of digits in \( n \) when it is written in base \( b \). Define the sequence \( f(n) \) by:

- \( f(1) = 1 \)
- \( f(2) = 2 \)
- \( f(n) = n \cdot f( d(n) ) \) for \( n > 2 \)

For which values of \( b \) does the series \( \frac{1}{f(1)} + \frac{1}{f(2)} + \frac{1}{f(3)} + \ldots \) converge?

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