
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 the sequence $\{a\_n\}$ is a geometric sequence with a common ratio $q=2$, $S\_n$ denotes the sum of the first $n$ terms of $\{a\_n\}$, and $T\_n = \frac{9S\_n - S_{2n}}{a_{n+1}} (n \in \mathbb{N}^*)$. Determine the maximum value of the sequence $\{T\_n\}$.

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