
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
Let the sequence $\{a_n\}$ have a sum of the first $n$ terms denoted as $S_n$, and for any positive integer $n$, the relation $a_n=5S_n+1$ holds. Define another sequence $\{b_n\}$ where $b_n=-1-\log_2|a_n|$, and let the sum of the first $n$ terms of $\{b_n\}$ be $T_n$. Define $c_n= \frac {b_{n+1}}{T_nT_{n+1}}$.  
$(1)$ Find the general formula for the sequence $\{a_n\}$ and the sum of the first $n$ terms of $\{c_n\}$, denoted as $A_n$;  
$(2)$ For any positive integers $m$ and $k$, does there exist an element $a_n$ in the sequence $\{a_n\}$ such that $|S_m-S_k|\leqslant 32a_n$ holds? If it exists, find the set of values for the positive integer $n$; if not, explain why.

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