
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
Let  $n$  be a positive integer. We are given a regular  $4n$ -gon in the plane. We divide its vertices in  $2n$  pairs and connect the two vertices in each pair by a line segment. What is the maximal possible amount of distinct intersections of the segments?

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