
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

On a circle, \(2N\) points are marked (\(N\) is a natural number). It is known that no more than two chords with endpoints at the 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 an endpoint of exactly one chord. 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 matchings and odd matchings.

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