
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 both $a\_n$ and $b\_n$ are sequences where the first term is 2 and all terms are positive numbers. $a\_n$ is a geometric sequence and $b\_n$ is an arithmetic sequence. It is known that $b\_2 = 4a\_2$ and $a\_2b\_3 = 6$.

1. Find the general term formulas for the sequences $a\_n$ and $b\_n$.
2. Find the smallest positive integer $n$ such that $a_{b\_n} < 0.001$.

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