
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, \(2n-1 \, (n \geq 3)\) different points are given. Find the minimal natural number \(N\) with the property that whenever \(N\) of the given points are colored black, there exist two black points such that the interior of one of the corresponding arcs contains exactly \(n\) of the given \(2n-1\) points.

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