
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
In a given regular $n$-gon with $n \geq 10$, select $k$ points from the $n$ vertices such that there exist 4 points among the $k$ points which form the vertices of a quadrilateral. This quadrilateral must have three of its sides as the sides of the given regular $n$-gon. Find the minimum value of $k$.

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