
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 $\{a_n\}$, the sum of the first $n$ terms $S_n$ satisfies $a_{n+1}=2S_n+6$, and $a_1=6$.

(Ⅰ) Find the general formula for the sequence $\{a_n\}$;

(Ⅱ) Let $b_n=\frac{a_n}{(a_n-2)(a_{n+1}-2)}$, and $T_n$ be the sum of the first $n$ terms of the sequence $\{b_n\}$. Is there a maximum integer $m$ such that for any $n\in \mathbb{N}^*$, $T_n > \frac{m}{16}$ holds? If it exists, find $m$; if not, explain why.

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