
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
$n$ points are located at the vertices of a convex $n$-gon. Inside this $n$-gon, $k$ additional points are marked. It turns out that any three of the $n+k$ points do not lie on the same line and form the vertices of an isosceles triangle. What can be the value of $k$?

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