
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 $\{a_n\}$ that satisfies $a_3 - 2a_6^2 + 3a_7 = 0$, and a geometric sequence $\{b_n\}$ for which $b_6 = a_6$, find the value of $b_1b_7b_{10}$.

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