
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 two sequences $\{a_n\}$ and $\{b_n\}$ satisfy $a_1 \cdot a_2 \cdot a_3 \cdots a_n = (\sqrt{2})^{b_n}$ (where $n \in \mathbb{N}^*$). If $\{a_n\}$ is a geometric sequence with all 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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