Suppose that \(C\) is a convex subset of \(\mathbb{R}^3\) with positive volume. Suppose that \(C_1, C_2,\dots C_n\) are \(n\) translated (not rotated) copies of \(C\) such that \(C_i \cap C \neq 0\), but \(C_i\) and \(C_j\) intersect at most on the boundary for \(i\neq j\). Prove that \(n \leq 27\) and that 27 is the best bound.

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