
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 \( p(n) \) denote the product of the non-zero digits of \( n \) (if \( n \) has only one digit, then \( p(n) \) equals that digit). If \( S = p(1) + p(2) + p(3) + \cdots + p(999) \), what is the greatest prime factor of \( S \)?

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