
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
Suppose the sequence $\{a_n\}$ is an increasing arithmetic sequence, where $a_3=5$, and $(a_1,a_2,a_5)$ forms a geometric sequence.
(1) Find the general term formula for $\{a_n\}$;
(2) Let $b_n=\frac{1}{(a_n+1)(a_{n+1}+1)}$, find the sum of the first $n$ terms $T_n$ of the sequence $\{b_n\}$;
(3) Does there exist a natural number $m$ such that $\frac{m-2}{4} < T_n < \frac{m}{5}$ holds for all $n (∈\mathbb{N}^*)$? If so, find the value of $m$; if not, explain the reason.

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