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Given graphs $G$ and $H$ with $|V(H)|=k$, the lexicographic product $G[H]$ is the graph obtained by substituting every vertex of $G$ by a copy of $H$, and every edge of $G$ by a copy of $K_{k,k}$. %Our main results are the following. We prove that for a sufficiently long path $P$, a nonrepetitive coloring of $P[K_k]$ needs at least $3k+\\lfloor k/2\\rfloor$ colors. 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