
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  $n\geq 3$  be an integer. We say that a vertex  $A_i (1\leq i\leq n)$  of a convex polygon  $A_1A_2 \dots A_n$  is *Bohemian* if its reflection with respect to the midpoint of  $A_{i-1}A_{i+1}$  (with  $A_0=A_n$  and  $A_1=A_{n+1}$ ) lies inside or on the boundary of the polygon  $A_1A_2\dots A_n$ . Determine the smallest possible number of Bohemian  vertices a convex  $n$ -gon can have (depending on  $n$ ).

*Proposed by Dominik Burek, Poland*

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