
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
Given a convex $n$-gon $A_{1} \ldots A_{n}$. Let $P_{i}(i=1, \ldots, n)$ be points on its boundary such that the line $A_{i} P_{i}$ divides its area in half. It is known that all points $P_{i}$ do not coincide with the vertices and lie on $k$ sides of the $n$-gon. What is a) the smallest; b) the greatest possible value of $k$ for any given $n$?

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