  We calculate various azimuthal angle distributions for three jets produced in
the forward rapidity region with transverse momenta $p_T>20\,\mathrm{GeV}$ in
proton-proton (p-p) and proton-lead (p-Pb) collisions at center of mass energy
$5.02\,\,\mathrm{TeV}$. We use the multi-parton extension of the so-called
small-$x$ Improved Transverse Momentum Dependent factorization (ITMD). We study
effects related to change from the standard $k_T$-factorization to ITMD
factorization as well as changes as one goes from p-p collision to p-Pb. We
observe rather large differences in the distribution when we change the
factorization approach, which allows to both improve the small-$x$ TMD gluon
distributions as well as validate and improve the factorization approach. We
also see significant depletion of the nuclear modification ratio, indicating a
possibility of searches for saturation effects using trijet final states in a
more exclusive way than for dijets.
