
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

Given positive integers \( n \) and \( k \), where \( n \) is even, \( k \geq 2 \), and \( n > 4k \), there are \( n \) points on the circumference of a circle. A set of \(\frac{n}{2}\) chords within the circle is called a "matching" if the endpoints of these chords are exactly the \( n \) points and the chords do not intersect each other inside the circle. Determine the largest integer \( m \) such that for any matching, there exist \( k \) consecutive points on the circumference such that at least \( m \) chords have both endpoints among these \( k \) points.

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