
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
Take \( n \geq 3 \) and consider a set \( E \) of \( 2n-1 \) distinct points on a circle. Suppose that exactly \( k \) of these points are to be colored black. Such a coloring is "good" if there is at least one pair of black points such that the interior of one of the arcs between them contains exactly \( n \) points from \( E \). Find the smallest value of \( k \) so that every such coloring of \( k \) points of \( E \) is good.

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