  We extend the previously developed small $x$ parton shower algorithm to
include the kinematic constraint effect and $k_t$ resummation effect. This work
enables the Monte Carlo generator to simultaneously resum large $k_t$ and small
$x$ logarithms in the saturation regime for the first time. It is an important
step towards simulating processes involving multiple well separated hard
scales, such as di-jet production in eA collisions at EIC.
