Let $A_1, A_2, \ldots, A_n$ be the vertices of a regular polygon with $n$ sides centered at the origin. Let $\vec{a}_k$ be the position vector of the point $A_k$ for $k = 1, 2, \ldots, n$. If $\left|\sum_{k=1}^{n-1} (\vec{a}_k \times \vec{a}_{k+1} + 1)\right| = \left|\sum_{k=1}^{n-1} (\vec{a}_k \cdot \vec{a}_{k+1} + 1)\right|$, then the minimum value of $n$ is