
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 $\{a_n\}$ with the first term being $2$ and the sum of the first $n$ terms being $S_n$, and $\frac{1}{a_n} - \frac{1}{a_{n+1}} = \frac{2}{4S_n-1} (n \in \mathbb{N}^*)$.

$(1)$ Find the value of $a_2$;  
$(2)$ Let $b_n = \frac{a_n}{a_{n+1}-a_n}$, find the general formula for the sequence $\{b_n\}$;  
$(3)$ Determine whether there exists a positive integer $n$ such that $\frac{a_{n+3}}{a_n}$ is an integer. If it exists, find $n$; if not, explain why.

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