Musical Chairs

$N$ people play a game of musical chairs — with a bit of a twist. There are $N$ chairs arranged in a circle in clockwise order, numbered $1, 2, 3, \ldots, N$. For each $1 \leq i \lt N$, chair $i$ is followed by chair $(i+1)$, and chair $N$ is followed by chair $1$. Initially, person $i$ stands in front of chair $i$. Further, chair $i$ has color $C_i$, and person $i$ is given a *banned* colo