
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
Each positive integer number  $n \ ge 1$  is assigned the number  $p_n$  which is the product of all its non-zero digits. For example,  $p_6 = 6$ ,  $p_ {32} = 6$ ,  $p_ {203} = 6$ . Let  $S = p_1 + p_2 + p_3 + \dots + p_ {999}$ . Find the largest prime that divides  $S $ .

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