
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 sum of the first $n$ terms ($S_n$) of the sequence $\{a_n\}$ satisfies $S_n = 2a_n - 1$ ($n \in \mathbb{N}^*$).

(1) Find the general term formula of the sequence $\{a_n\}$;

(2) If the sequence $\{b_n\}$ satisfies $b_n = 1 + \log_2 a_n$,

(I) Find the sum of the first $n$ terms ($T_n$) of the sequence $\{a_n b_n\}$;

(II) Find the minimum value of $\frac{b_n^2 + 9}{(\log_2 a_n) + 2}$.

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