
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
Denote by  $P(n)$  the product of the digits of a positive integer  $n$ . For example,  $P(1948)=1\cdot9\cdot4\cdot8=288$ .

[list=a]
[*] Evaluate the sum  $P(1)+P(2)+\dots+P(2017)$ .
[*] Determine the maximum value of  $\frac{P(n)}{n}$  where  $2017\leq n\leq5777$ .
[/list]

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