Let $ \mathcal{H} = A_1A_2\ldots A_n$ be a convex $ n$-gon. For $ i = 1, 2, \ldots, n$, let $ A'_{i}$ be the point symmetric to $ A_i$ with respect to the midpoint of $ A_{i - 1}A_{i + 1}$ (where $ A_{n + 1} = A_1$). We say that the vertex $ A_i$ is good if $ A'_{i}$ lies inside $ \mathcal{H}$. Show that at least $ n - 3$ vertices of $ \mathcal{H}$ are good.