
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 $2$, and the sum of the first $n$ terms $S_n$. It is known that $\frac{1}{a_n}-\frac{1}{a_{n+1}}=\frac{2}{4S_n-1}$ for $n\in\mathbb{N}^*$.

(1) Find the value of $a_2$;
(2) If $\{b_n=\frac{a_n}{a_{n+1}-a_n}\}$, find the general formula for the sequence $\{b_n\}$;
(3) Find the general formula for the sequence $\{a_n\}$.

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