
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 $\{a\_n\}$ is a geometric sequence, with the first term $a\_1=1$, common ratio $q > 0$, and the sum of the first $n$ terms denoted as $S\_n$. It is also known that $S\_1+a\_1$, $S\_3+a\_3$, $S\_2+a\_2$ form an arithmetic sequence.

(1) Find the general term formula for the sequence $\{a\_n\}$;

(2) If the sequences $\{b\_n\}$ and $\{c\_n\}$ satisfy $\frac{b\_n}{n+2}=-\log\_2 a\_{n+1}$, and $b\_n \cdot c\_n = 1$, let $T\_n$ denote the sum of the first $n$ terms of the sequence $\{c\_n\}$. If $T\_n \geq m$ always holds true, find the maximum value of $m$.

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