In this article, we consider the problem of finding a zero of a system of monotone inclusions in Hilbert spaces. Notably, each of these monotone inclusions comprises three operators, with two of them being linearly composed. To address this challenge, we propose a new splitting method that, at each iteration, essentially necessitates the computation of three individual resolvents, corresponding to each operator within the monotone inclusion. Under the weakest possible conditions, with the help of characteristic operator techniques, we analyze the weak convergence properties of our proposed method, which is facilitated by the introduction of a novel inequality. Numerical results demonstrate the practical usefulness of this method in solving large-scale rare feature selection in deep learning.