
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  $\mathbb{Z}_{\ge 0}$  denote the set of nonnegative integers.

Define a function  $f:\mathbb{Z}_{\ge 0} \to\mathbb{Z}$  with  $f\left(0\right)=1$  and \[ f\left(n\right)=512^{\left\lfloor n/10 \right\rfloor}f\left(\left\lfloor n/10 \right\rfloor\right)\]
for all  $n \ge 1$ . Determine the number of nonnegative integers  $n$  such that the hexadecimal (base  $16$ ) representation of  $f\left(n\right)$  contains no more than  $2500$  digits.

*Proposed by Tristan Shin*

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