
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 points \( A_{1}, A_{2}, \ldots, A_{2n} \) are the vertices of a regular polygon with \( 2n \) sides. Find the number of triplets \( A_{i}, A_{j}, A_{k} \) such that the triangle \( A_{i} A_{j} A_{k} \) is right-angled and the number of triplets such that the triangle is acute-angled.

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