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We classify the tilting complexes $T$ in $D^b(\\operatorname{coh}\\mathbb{X})$ such that $\\tau^2 T\\cong T$, where $\\tau$ is the Auslander-Reiten translation in $D^b(\\operatorname{coh}\\mathbb{X})$. As an application of this result, we classify the 2-representation-finite algebras which are derived-equivalent to a canonical algebra. This complements Iyama-Oppermann's classification of the iterated tilted 2-representation-finite algebras. 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