
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 a sequence $\{a\_n\}$ where all terms are greater than $1$, the sum of the first $n$ terms is $S\_n$, and it satisfies $a\_n^{2}+3a\_n=6S\_n-2$.

1. Find the general term formula for the sequence $\{a\_n\}$.
2. Let $b\_n= \frac {1}{a\_na\_{n+1}}$, find the sum of the first $n$ terms, $T\_n$, for the sequence $\{b\_n\}$.
3. Find the smallest positive integer $m$ such that $T\_n < \frac {m}{36}$ holds for all $n∈N^{*}$.

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