
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
On the extensions of the sides \(A_{1} A_{2}, A_{2} A_{3}, \ldots, A_{n} A_{1}\) of a regular \(n\)-gon (\(n \geqslant 5\)), construct points \(B_{1}, B_{2}, \ldots, B_{n}\) such that \(B_{1} B_{2}\) is perpendicular to \(A_{1} A_{2}\), \(B_{2} B_{3}\) is perpendicular to \(A_{2} A_{3}\), and so on, with \(B_{n} B_{1}\) being perpendicular to \(A_{n} A_{1}\).

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