
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 that $S_{n}$ is the sum of the first $n$ terms of the sequence $\{a_{n}\}$, and $a_{1}=1$, $S_{n}+1=a_{n+1}$ $(n\in\mathbb{N}^{*})$.

$(1)$ Find the general term $a_{n}$ of the sequence $\{a_{n}\}$;  
$(2)$ If $b_{n}= \frac {n}{4a_{n}}$, find the sum of the first $n$ terms $T_{n}$ of the sequence $\{b_{n}\}$;  
$(3)$ Let $c_{k}= \frac {k+2}{S_{k}(T_{k}+k+1)}$, the sum of the first $n$ terms of $\{c_{k}\}$ is $A_{n}$. Is there a minimum positive integer $m$ such that the inequality $A_{n} < m$ holds for any positive integer $n$? 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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