
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
There are $2N$ points marked on a circle ($N$ is a natural number). It is known that through any point inside the circle, no more than two chords with ends at the marked points pass. We define a matching as a set of $N$ chords with ends at the marked points such that each marked point is an end of exactly one of these chords. A matching is called even if the number of intersection points of its chords is even, and odd otherwise. Find the difference between the number of even and odd matchings.

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