
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 a non-zero arithmetic sequence $\left\{ a_n \right\}$ where $2a_3-a_7^2+2a_{11}=0$, and a geometric sequence $\left\{ b_n \right\}$ with $b_7=a_7$, find the product of $b_6$ and $b_8$.

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