
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 $2a_2 + 2a_{12} = a_7^2$, and a geometric sequence $\{b_n\}$ with $b_7 = a_7$, find the value of $b_5b_9$.

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