
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
Fix an integer  $ b \geq 2$ . Let  $ f(1) \equal{} 1$ ,  $ f(2) \equal{} 2$ , and for each  $ n \geq 3$ , define  $ f(n) \equal{} n f(d)$ , where  $ d$  is the number of base- $ b$  digits of  $ n$ . For which values of  $ b$  does
\[ \sum_{n\equal{}1}^\infty \frac{1}{f(n)} 
\]
converge?

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