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We give a formula for the crystal structure on the integer points of thestring polytopes and the $*$-crystal structure on the integer points of thestring cones of type $A$ for arbitrary reduced words. As a byproduct we obtaindefining inequalities for Nakashima-Zelevinsky string polytopes. Furthermore,we give an explicit description of the Kashiwara $*$-involution on string datafor a special choice of reduced word.

Combinatorics of canonical bases revisited: String data in type $A$