
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 $a_1a_2\cdots a_n=(\sqrt{2})^{b_n}$, where $n\in\mathbb{N}^*$, if $\{a_n\}$ is a geometric sequence and $a_1=2$, $b_3=6+b_2$,

$(1)$ find $a_3$ and the general formula for the sequence $\{b_n\}$;

$(2)$ let $c_n=\frac{1}{a_n}-\frac{1}{b_n}$, where $n\in\mathbb{N}^*$, and denote the sum of the first $n$ terms of the sequence $\{c_n\}$ as $S_n$.

    $(i)$ find $S_n$;

    $(ii)$ if $S_k\geqslant S_n$ always holds, find the value of the positive integer $k$.

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