
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 it satisfies $a_1=1$, $S_{n}+1=a_{n+1}$ $(n \in \mathbb{N}^{*})$.  
(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 $c_k=\frac{k+2}{S_k(T_k+k+1)}$, and the sum of the first $n$ terms of $\{c_k\}$ be $A_n$. 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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