
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 non-zero function defined on $\mathbb{R}$, for any $x, y \in \mathbb{R}$, the relationship $f(xy) = xf(y) + yf(x)$ holds. The sequence $\{a_n\}$ satisfies $a_n = f(2^n)$ for $n \in \mathbb{N}^*$, and $a_1 = 2$. Find the general term formula for the sequence $\{a_n\}$, $a_n = $ _________.

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