
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
Let the sum of the first n terms of the sequence {a<sub>n</sub>} be S<sub>n</sub>, _____．  
Choose one of the following three conditions to supplement the question below and answer it:  
① The sequence {a<sub>n</sub>} is a geometric sequence with a common ratio of 2, and a<sub>2</sub>, a<sub>3</sub>, a<sub>4</sub>-4 form an arithmetic sequence;  
② S<sub>n</sub> = 2a<sub>n</sub> - 2;  
③ S<sub>n</sub> = 2<sup>n+1</sup> - 2.  

(1) Find the general formula for the sequence {a<sub>n</sub>};  
(2) If b<sub>n</sub> = $\frac{{1+{{log}_2}{a_n}}}{{{a_n}}}$, find the sum of the first n terms of the sequence {b<sub>n</sub>, T<sub>n</sub>}.

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