
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 $\{a_n\}$ whose sum of the first $n$ terms is $s_n$, and $a_n=S_{n-1}+2$ $(n\geqslant 2)$, $a_1=2$.  
$(1)$ Find the general formula for the sequence $\{a_n\}$.  
$(2)$ Let $b_n= \frac {1}{\log _{2}a_n}$, $T_n=b_{n+1}+b_{n+2}+\ldots+b_{2n}$. Is there a maximum positive integer $k$ such that for any positive integer $n$, $T_n > \frac {k}{12}$ always holds?  
If it exists, find the value of $k$; if not, explain why.

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