
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
In a sequence, insert the product of every two adjacent terms between them to form a new sequence, which is called an "expansion" of the sequence. Expanding the sequence $1$, $2$, the first time we get $1$, $2$, $2$; the second time we get $1$, $2$, $2$, $4$, $2$; ... Suppose that after the $n$-th "expansion", the obtained sequence is $1$, $x\_1$, $x\_2$, ..., $x\_m$, $2$, and let $a\_n = \log\_2(1 \cdot x\_1 \cdot x\_2 \cdot \ldots \cdot x\_m \cdot 2)$, then the general term formula of the sequence $\{a\_n\}$ is $\_\_\_\_\_\_\_$.

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