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