
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 sequence $\{a_n\}$ has a sum of the first $n$ terms denoted as $S_n$, with the first term being $a\_1$, and $\frac{1}{2}$, $a_n$, $S_n$ form an arithmetic sequence.

(1) Find the general formula for the $n$-th term of the sequence $\{a_n\}$.

(2) The sequence $\{b_n\}$ is defined by $b_n = (\log_2 a_{2n+1}) \times (\log_2 a_{2n+3})$. Find the sum of the first $n$ terms of the sequence $\left\{ \frac{1}{b_n} \right\}$, denoted as $T_n$.

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