
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 the sequences $\{a\_n\}$ and $\{b\_n\}$ that satisfy $a\_1 \cdot a\_2 \cdot a\_3 \dots a\_n = (\sqrt{2})^{b\_n} (n \in \mathbb{N}^*)$. If $\{a\_n\}$ is a geometric sequence with positive terms, and $a\_1 = 2$, $b\_3 = 6 + b\_2$, find the expression for $b\_n =$ \_\_\_\_\_\_.

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