
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
In an arithmetic sequence $\{a_n\}$ with a non-zero common difference, it is given that $3a_{2005} - a_{2007}^2 + 3a_{2009} = 0$. The sequence $\{b_n\}$ is a geometric sequence, and $b_{2007} = a_{2007}$. Find the value of $b_{2006}b_{2008}$.

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