  Lascoux polynomials generalize Grassmannian stable Grothendieck polynomials
and may be viewed as K-theoretic analogs of key polynomials. The latter two
polynomials have combinatorial formulas involving tableaux: Lascoux and
Sch\"{u}tzenberger gave a combinatorial formula for key polynomials using right
keys; Buch gave a set-valued tableau formula for Grassmannian stable
Grothendieck polynomials. We establish a novel combinatorial rule for Lascoux
polynomials involving right keys and set-valued tableaux. Our rule recovers the
tableaux formulas of key polynomials and Grassmannian stable Grothendieck
polynomials. To prove our rule, we construct a new abstract Kashiwara crystal
structure on set-valued tableaux. This construction answers an open problem of
Monical, Pechenik and Scrimshaw in the context of abstract Kashiwara crystal.
