
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 geometric sequence {$a_n$} with the sum of the first n terms being $S_n$, and that $2^{n+1}$, $S_n$, and a form an arithmetic sequence (where $n \in \mathbb{N}^*$),
(1) Determine the value of a and the general term formula for the sequence {$a_n$};
(2) If $b_n = (1 - a_n)\log_2(a_n a_{n+1})$, find the sum of the first n terms, $T_n$, of the sequence {$\frac{1}{b_n}$}.

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