
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 \((n \geq 5)\) \( A_{1} A_{2} \ldots A_{n} \), 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 \( A_{2} A_{3} \), \(\ldots, B_{n} B_{1} \) is perpendicular to \( A_{n} A_{1} \).

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