
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 that $f(x)$ is a function defined on $\mathbb{R}$ that is not always zero, and for any $x, y \in \mathbb{R}$, $f(x \cdot y) = xf(y) + yf(x)$ holds. The sequence $\{a\_n\}$ satisfies $a\_n = f(3^n) (n \in \mathbb{N}_+)$, and $a\_1 = 3$. Find the general term formula for the sequence.

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