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

\\((1)\\) Find the general formula for 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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