
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 that the sequence $\{a_{n}\}$ is an arithmetic sequence, and the sequence $\{b_{n}\}$ is a geometric sequence, $\frac{{a}_{2}+{a}_{8}}{1+{b}_{2}{b}_{8}}$ can be expressed as $\frac{a_{1}+d+a_{1}+7d}{1+b_{2}^{2}b_{8}^{2}}$, where $a_{1}$ and $d$ are the first term and common difference of the arithmetic sequence, and $b_{2}$ and $b_{8}$ are the second and eighth terms of the geometric sequence, respectively. Given that ${a}_{1}+{a}_{5}+{a}_{9}=9$ and ${b}_{2}{b}_{5}{b}_{8}=3\sqrt{3}$, calculate $\frac{{a}_{2}+{a}_{8}}{1+{b}_{2}{b}_{8}}$.

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