
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
A line cuts off a triangle \( APQ \) from a regular \( n \)-gon with side length 1 such that \( AP + AQ = 1 \) (\( A \) is a vertex of the \( n \)-gon).
Find the sum of the angles at which the line segment \( PQ \) is visible from all vertices of the \( n \)-gon, except \( A \).

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