
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 an arithmetic sequence $\{a_n\}$ where each term is non-zero and satisfies $a_3 - \frac{a_7^2}{2} + a_{11} = 0$, and a geometric sequence $\{b_n\}$ with $b_7 = a_7$, calculate $b_1 \cdot b_{13}$.

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