
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
Given a positive integer  $n$ , let  $p(n)$  be the product of the non-zero digits of  $n$ .  (If  $n$  has only one digits, then  $p(n)$  is equal to that digit.)  Let \[ S=p(1)+p(2)+p(3)+\cdots+p(999). \]  What is the largest prime factor of  $S$ ?

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