Let $A_1A_2\cdots A_n$ be a regular polygon with $n$ sides, $n \geqslant 3$. Initially, there are three ants standing at vertex $A_1$. Every minute, two ants simultaneously move to an adjacent vertex, but in different directions (one clockwise and the other counterclockwise), and the third stays at its current vertex. Determine all the values of $n$ for which, after some time, the three ants can meet at the same vertex of the polygon, different from $A_1$.