
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
Let \( P(n) \) denote the product of the digits of a natural number \( n \). What is the largest natural number \( k \) for which there exists a natural number \( n > 10 \) such that

\[ P(n) < P(2n) < \ldots < P(kn) ? \]

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