
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
The side length of regular octagon \(A_{1} A_{2} A_{3} A_{4} A_{5} A_{6} A_{7} A_{8}\) is 1. If any two points \(A_{i}\) and \(A_{j}\) are taken, what is the maximum value of \(\overrightarrow{A_{i} A_{j}} \cdot \overrightarrow{A_{1} A_{2}}\)?

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