
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
Let $S\_n$ be the sum of the first $n$ terms of an arithmetic sequence $\{a\_n\}$ with common difference $d$. Then the sequence $S\_6 - S\_3$, $S\_9 - S\_6$, $S\_{12} - S\_9$ is an arithmetic sequence with common difference $9d$. By analogy, we can draw the conclusion that, if $T\_n$ is the product of the first $n$ terms of a geometric sequence $\{b\_n\}$ with common ratio $2$, then the sequence $\frac{T\_6}{T\_3}$, $\frac{T\_9}{T\_6}$, $\frac{T\_{12}}{T\_9}$ is a geometric sequence with common ratio $\_\_\_\_\_\_.$

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