
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 sum of the first $n$ terms of a positive sequence ${a_n}$ is $S_n$, and it satisfies: $$a_{3}=7，a_{n+1}^{2}=6S_{n}+9n+1，n∈N^{*}$$.
(1) Find the general term formula for the sequence ${a_n}$.
(2) If the positive geometric sequence ${b_n}$ satisfies $b_{1}=a_{1}$, $b_{3}=a_{2}$, and $c_{n}=a_{n}•b_{n}$, with the sum of the first $n$ terms of the sequence ${c_{n}}$ denoted as $T_{n}$, and for any $n≥2$, $n∈N^{*}$, the inequality $$(T_{n}-5)m≥6n^{2}-31n+35$$ always holds, find the range of the real number $m$.

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