
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
By generalizing the basic inequality $\sqrt{ab}≤\frac{a+b}{2}(a＞0,b＞0)$, we can derive the correct conclusion $\sqrt[3]{abc}≤\frac{a+b+c}{3}(a＞0,b＞0,c＞0)$, where the equality holds if and only if $a=b=c$. Using this conclusion to solve the following problem: Given a rectangle with a perimeter of $18cm$, when the length of the rectangle is ______ $cm$, rotating it around one side forms a cylinder. The value of the volume of the cylinder when it is at its maximum is ______ $cm^{3}$.

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