
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, $S\_n$, satisfies $S\_n > 1$ and $6S\_n=(a\_n+1)(a\_n+2)$, $n∈N^{*}$.
(1) Find the general term formula of the sequence $\{a\_n\}$;
(2) If the sum of the first $n$ terms of the sequence $\{b\_n\}$ is $b\_n=−a\_n+19$, find the sum of the first $n$ terms of the sequence $\{|b\_n|\}$, $T\_n$.

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