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We prove a higher-dimensional generalization conjectured by Hassett and Tschinkel: for a holomorphic symplectic variety $M$ deformation equivalent to a Hilbert scheme of $n$ points on a K3 surface, an extremal curve class $R\\in H_2(M,\\mathbb{Z})$ in the Mori cone is the line in a Lagrangian $n$-plane $\\mathbb{P}^n\\subset M$ if and only if certain intersection-theoretic criteria are met. 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