Example 8 A diagonal of a 2006-gon $P$ is called good if its endpoints divide the boundary of $P$ into two parts, each containing an odd number of sides of $P$. In particular, a side of $P$ is also considered good.

Suppose $P$ is triangulated by 2003 diagonals that do not intersect in the interior of $P$. Find the maximum number of isosceles triangles in this triangulation that have two good sides. (47th IMO Problem)