
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 an arithmetic-geometric sequence $\{a\_n\}$ with the sum of the first $n$ terms denoted as $S\_n$, and it satisfies the conditions $S\_3 = 3a\_3 + 2a\_2$, $a\_4=8$.

(1) Find the general term formula for the sequence $\{a\_n\}$.

(2) Let $b\_n = \log\_2 a\_n$. Denote the sum of the first $n$ terms of the sequence $\{b\_n\}$ as $T\_n$. Find the positive integer value of $n$ that maximizes $T\_n$.

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