  A novel observable, the double nuclear modification factor, is proposed to
probe simultaneously the initial and final state effects in nucleus-nucleus
collisions. An interesting competition between the combinatorial enhancement in
the double parton scattering and the suppression due to parton energy loss can
be observed in the production rate of two hard particles. In particular, the
production of $J/\psi$ mesons in association with a $W$ boson is not suppressed
but is enhanced in the region of moderate transverse momenta, unlike the case
of unassociated (inclusive) $J/\psi$ production. At the same time, in the
region of high enough transverse momenta the nuclear modification factor for
associated $J/\psi+W$ production converges to that of unassociated $J/\psi$.
