
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 two sequences $\{a_n\}$ and $\{b_n\}$ satisfying $a_1a_2a_3…a_n=(\sqrt{2})^{b_n}$ $(n\in\mathbb{N}^*)$. If $\{a_n\}$ is a geometric sequence, and $a_1=2$, $b_3=6+b_2$.  
(I) Find $a_n$ and $b_n$;  
(II) Let $c_n= \frac{1}{a_n}- \frac{1}{b_n}$ $(n\in\mathbb{N}^*)$. Denote the sum of the first $n$ terms of the sequence $\{c_n\}$ as $S_n$.  
  (i) Find $S_n$;  
 (ii) Find the positive integer $k$, such that for any $n\in\mathbb{N}^*$, $S_k\geqslant S_n$.

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