
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
A sequence $\{a\_n\}$ is given, where $a\_1=8$, $a\_4=2$, and it satisfies $a\_{n+2}-2a\_{n+1}+a\_n=0 (n\in\mathbb{N}^*)$.

1. Find the general term formula for the sequence $\{a\_n\}$.
2. Let $b\_n= \frac {1}{n(12-a\_n)}(n\in\mathbb{N}^*)$, and $S\_n=b\_1+b\_2+…+b\_n$, does there exist a maximum integer $m$ such that for any $n$, $S\_n > \frac {m}{32}$ always holds? If it exists, find $m$; if not, explain the reason.

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