
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
Consider an geometric sequence $\{a_n\}$ with a common ratio $q=2$. Let $S_n$ denote the sum of the first $n$ terms of $\{a_n\}$, and define $T_n = \frac{9S_n - S_{2n}}{a_{n+1}}$ (where $n \in \mathbb{N}^*$). Find the maximum value of the sequence $\{T_n\}$.

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