
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, its sum of the first $n$ terms is $S\_n$, and $a\_2⋅a\_3=15$, $S\_4=16$.
(I) Find the general term formula for the sequence $\{a\_n\}$;
(II) The sequence $\{b\_n\}$ satisfies $b\_1=a\_1$, $b_{n+1}-b\_n= \frac {1}{a\_n\cdot a_{n+1}}$.
1. Find the general term formula for the sequence $\{b\_n\}$;
2. Does there exist a positive integer $m$, $n(m\neq n)$, such that $b\_2$, $b\_m$, $b\_n$ form an arithmetic sequence? If so, find the values of $m$ and $n$; if not, explain the reason.

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