
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
Let \( n \geq 3 \). Consider a set \( E \) of \( 2n - 1 \) distinct points on a circle. If some of the points in \( E \) are colored black in such a way that there are at least two black points, and for these two black points, one of the two arcs connecting them contains exactly \( n \) points from \( E \) in its interior, then this coloring method is called "good". If every coloring method where \( k \) points of \( E \) are colored black is "good", find the minimum value of \( k \).

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