Three coins, \(A, B, C\) are situated one at each vertex of an equilateral triangle of side \(n\). The triangle is divided into small equilateral triangles of side 1 by lines parallel to the sides. Initially, all the lines of the figure are blue. The coins move along the lines, painting in red their trajectory, following the two rules:

(i) First coin to move is \(A\), then \(B\), then \(C\), then again \(A\), and so on. At each turn, a coin paints exactly one side of one of the small triangles.

(ii) A coin can not move along a segment that is already painted red, but it can stay at an endpoint of a red segment, not necessarily alone.

Show that for all integers \(n>0\) it is possible to paint all the sides of all the small triangles red.