
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 function $f(n) = (2n+7) \cdot 3^n + 9$,
1. Calculate the value of $f(1)f(2)f(3)$.
2. Determine whether there exists an integer $m \geq 2$ such that for any positive integer $n$, $f(n)$ is divisible by $m$. If such an $m$ exists, find the largest possible value; if not, provide the reasoning.

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