
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 the sequences $\{a_n\}$ and $\{b_n\}$ that satisfy $(a_1a_2...a_n=(\sqrt{2})^{b_n}(n∈N^∗)).$ If $\{a_n\}$ is a geometric sequence, with $a_1=2, b_3=6+b_2,$ let $(C_n=\frac{1}{a_n}-\frac{1}{b_n}(n∈N^∗)),$ and denote the sum of the first $n$ terms of the sequence $\{C_n\}$ as $(S_n).$ If $k$ is a positive integer such that $(S_k \geq S_n)$ holds for any $(n∈N^∗)$, then $k=$_______.

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