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.
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