
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
For the sequence $\{a_n\}$, where $a_1=8$, $a_4=2$, and it satisfies $a_{n+2}=2a_{n+1}-a_n$ for $n\in\mathbb{N}^*$,  
(1) Find the general formula for the sequence $\{a_n\}$.  
(2) Let $S_n=|a_1|+|a_2|+\ldots+|a_n|$, find $S_n$.  
(3) Let $b_n= \frac {1}{n(12-a_n)}$ for $n\in\mathbb{N}^*$, and $T_n=b_1+b_2+\ldots+b_n$ for $n\in\mathbb{N}^*$, is there a maximum integer $m$ such that for any $n\in\mathbb{N}^*$, $T_n > \frac {m}{32}$ holds? If it exists, find the value of $m$; if not, explain why.

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