
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 the sum of the first $n$ terms is $S_n$, and $a_1=2$, $a_n=S_{n-1}+2$ ($n\geq2$).
1. Find the general term formula for the sequence $\{a_n\}$.
2. Let $b_n=\frac{1}{\log_2{a_n}}$, $T_n=b_{n+1}+b_{n+2}+\dots+b_{2n}$ ($n\in\mathbb{N}^*$). Does there exist a maximum positive integer $k$ such that $T_n>\frac{k}{12}$ always holds? If so, find the value of $k$; if not, explain the reason.

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