
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 all terms of the sequence $\{a\_n\}$ are positive numbers, and the sum of the first $n$ terms is $S\_n$. For any $n∈N^{*}$, it is known that $2S_{n}=n^{2}+n$.
(I) Find the general formula of the sequence $\{a\_n\}$;
(II) The sequence $\{b\_n\}$ satisfies $b_{1}=1$, $2b_{n+1}-b_{n}=0 (n∈N^{*})$, and $c_{n}=a_{n}b_{n}$. Find the sum of the first $n$ terms of the sequence $\{c\_n\}$, denoted as $T_{n}$;
(III) Under the condition of (II), determine whether there exists an integer $m$ such that for any positive integer $n$, $m-2 < T_{n} < m+2$ holds. If it exists, find the value of $m$; if not, explain the reason.

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