
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 an arithmetic sequence $\{a\_n\}$ with the sum of its first $n$ terms denoted as $S\_n$, where $a\_1=\frac{1}{2}$, common ratio $q > 0$, and the sequence formed by $S\_1+a\_1$, $S\_3+a\_3$, $S\_2+a\_2$ is an arithmetic sequence.

1. Find $a\_n$;
2. Let $b\_n=\frac{1}{(\log\_2a\_n)^2}$ and $c\_n=(n+1)b\_nb_{n+2}$. Find the sum of the first $n$ terms of the sequence $\{c\_n\}$, denoted as $T\_n$.

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