
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
For sequences $\{a_n\}$ and $\{b_n\}$, it is known that $a_1a_2a_3…a_n=2^{b_n}$ $(n\in\mathbb{N}^*)$, and $a_1=2$, $b_3-b_2=3$. If the sequence $\{a_n\}$ is a geometric sequence.  
(I) Find $a_3$ and the general formula for the sequence $\{b_n\}$.  
(II) Let $c_n= \frac {2b_n}{n^2}$, is there any positive integers $m$, $n$ ($m\neq n$), such that $c_2$, $c_m$, $c_n$ form an arithmetic sequence? If so, find the values of $m$ and $n$; if not, explain why.

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