In an island shaped like a regular polygon of $n$ sides, there are airports at each vertex of the island. The island would like to add $k$ new airports into the interior of the island, but it must follow the following rules: $1$. It must be in the interior of the island (none on borders). $2$. No two airports can be at the exact same location. $3$. Every triple of $1$ new and $2$ old airports must form an isoceles triangle. $4$. No three airports can be collinear. Find the maximum value of $k$ for each $n$

HIDE: Harder Version Replace $1$ new and $2$ old with $1$ old and $2$ new.