
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 $\{a_n\}$ where each term is a positive number and the sum of the first $n$ terms is $S_n$, with the first term being $a_1$, and given that $2$, $a_n$, and $S_n$ form an arithmetic sequence.

(1) Find the general formula for the sequence $\{a_n\}$;
(2) If $b_n = \log_2 a_n$ and $c_n = \frac{1}{b_n b_{n+1}}$, find the sum of the first $n$ terms, $T_n$, of the sequence $\{c_n\}$.

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