Octagons

Below is a picture of an infinite hyperbolic tessellation of octagons. If we think of this as a graph of vertices (of degree three), then there exists an isomorphism of the graph that maps any vertex $x$ onto any other vertex $y$. Every edge is given a label from the set \{a, b, c\} in such a way that every vertex has all three types of edges incident to it, and the labels alternate around each octagon. Part of this labeling is illustrated in the diagram.

So, a path in this graph (starting from any vertex) can be specified by a sequence of edge labels. Your task is to write a program which, given a sequence of labels such as $\text{abcbcbcabcaccabb}$, outputs $\text{closed}$ if the path ends at the same vertex where it started, and $\text{open}$ otherwise.