
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]


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Given that the sum of the first $n$ terms of a sequence of positive numbers $\{\_a\_n\}$ is $S\_n$, and $a\_n$ and $S\_n$ satisfy the equation $4S\_n=(a\_n+1)^2(n=1,2,3…)$.

(1) Find the general term formula of $\{\_a\_n\}$.

(2) If $b\_n=\frac{1}{a\_n\cdot a\_{n+1}}$, find the sum of the first $n$ terms of $\{\_b\_n\}$, denoted as $T\_n$.

(3) Under the conditions of (2), for any $n\in N^{*}$, if $T\_n>\frac{m}{23}$ holds true, find the maximum value of the integer $m$.

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