
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 that the sequence $\left\{a_{n}\right\}$ is an arithmetic sequence and $\left\{b_{n}\right\}$ is a geometric sequence, and that $a_{1}+2 b_{1}, a_{3}+4 b_{3}, a_{5}+8 b_{5}$ form an arithmetic sequence. Find $\frac{b_{3} b_{7}}{b_{4}^{2}}$.

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