
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 \( n \). For example, \( p(7)=7 \), \( p(204)=2 \times 4=8 \), etc. Let \( S=p(1)+p(2)+\cdots+p(999) \). What is the largest prime factor of \( S \)?

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