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The Box-Ball System (BBS) is a cellular automaton introduced by Takahashi and Satsuma in the 1990s. The system is a discrete counterpart of the KdV equation and exhibits solitonic behavior. Recently, the BBS started from a random two-sided infinite particle configuration has been studied, by encoding the particle configuration to a certain discrete path on Z and defining the dynamic of the BBS in terms of the path. In this paper, we extend some results of the previous study in this direction to a generalization of the BBS with \k{appa}-color balls (particles), called the multicolor BBS. We first introduce an encoding of the \k{appa}-color particles configuration to a discrete path in \k{appa}-dimensional Euclidean space. Then we show that the dynamics of the multicolor BBS is expressed by the composition of the Pitman's transformation and a permutation operator. Applying this expression, we characterize the set of configurations for which the dynamics are well-defined and reversible for all times. Then, we give a simple class of random initial conditions which are invariant in distribution under the dynamics of the multicolor BBS. Finally, we introduce a continuous version of the multicolor BBS, which is defined for continuous paths on R in \k{appa}-dimensional Euclidean space, and show that \k{appa}-dimensional Brownian motion with a proper drift is invariant under the dynamics of the continuous version of the multicolor BBS.