
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
For  $n \in \mathbb{N}$ , let  $P(n)$  denote the product of the digits in  $n$  and  $S(n)$  denote the sum of the digits in  $n$ . Consider the set $A=\{n \in \mathbb{N}: P(n)$  is non-zero, square free and  $S(n)$  is a proper divisor of  $P(n)\}$ .
Find the maximum possible number of digits of the numbers in  $A$ .

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