
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 a positive geometric sequence $\{a_n\}$ satisfying the conditions that $a_1$, $2a_2$, and $a_3 + 6$ form an arithmetic sequence, and $a_4^2 = 9a_1a_5$,

(I) Find the general term formula for the sequence $\{a_n\}$;

(II) Let $b_n = (\log_{\sqrt{3}}(a_n) + 1) \cdot a_n$, find the sum of the first $n$ terms $T_n$ of the sequence $\{b_n\}$.

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