  The invariant yields for $J/\psi$ production at forward rapidity
$(1.2<|y|<2.2)$ in U$+$U collisions at $\sqrt{s_{_{NN}}}$=193 GeV have been
measured as a function of collision centrality. The invariant yields and
nuclear-modification factor $R_{AA}$ are presented and compared with those from
Au$+$Au collisions in the same rapidity range. Additionally, the direct ratio
of the invariant yields from U$+$U and Au$+$Au collisions within the same
centrality class is presented, and used to investigate the role of $c\bar{c}$
coalescence. Two different parameterizations of the deformed Woods-Saxon
distribution were used in Glauber calculations to determine the values of the
number of nucleon-nucleon collisions in each centrality class, $N_{\rm coll}$,
and these were found to give significantly different $N_{\rm coll}$ values.
Results using $N_{\rm coll}$ values from both deformed Woods-Saxon
distributions are presented. The measured ratios show that the $J/\psi$
suppression, relative to binary collision scaling, is similar in U$+$U and
Au$+$Au for peripheral and midcentral collisions, but that $J/\psi$ show less
suppression for the most central U$+$U collisions. The results are consistent
with a picture in which, for central collisions, increase in the $J/\psi$ yield
due to $c\bar{c}$ coalescence becomes more important than the decrease in yield
due to increased energy density. For midcentral collisions, the conclusions
about the balance between $c\bar{c}$ coalescence and suppression depend on
which deformed Woods-Saxon distribution is used to determine $N_{\rm coll}$.
