
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
$2N$ points are marked on a circle ($N$ is a natural number). It is known that no more than two chords with endpoints at these marked points pass through any point inside the circle. A matching is defined as a set of $N$ chords with endpoints at the marked points such that each marked point is the endpoint of exactly one of these chords. We call a matching even if the number of points where its chords intersect 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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