
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 regular octagon \( A_{1} A_{2} A_{3} A_{4} A_{5} A_{6} A_{7} A_{8} \) has a side length of 1. If we select any two points \( A_i \) and \( A_j \), 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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