
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 the sequence $\{a_n\}$ is an arithmetic sequence with a positive common difference, and the sum of the first $n$ terms is $S_n$, and $a_2 \cdot a_3=15$, $S_4=16$.

(1) Find the general formula for the sequence $\{a_n\}$.

(2) The sequence $\{b_n\}$ satisfies $b_1=a_1$, $b_{n+1}-b_n=\frac{1}{a_n \cdot a_{n+1}}$.
   - (i) Find the general formula for the sequence $\{b_n\}$.
   - (ii) Determine whether there exist positive integers $m$, $n$ ($m \neq n$), such that $b_2$, $b_m$, $b_n$ form an arithmetic sequence.

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