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One crucial issue is the inexact computation of the proximal operator, often implemented through a nested primal\u2013dual solver, which represents the main computational bottleneck whenever an increasing accuracy in the computation is required. In this paper, we propose a nested primal\u2013dual method for the efficient solution of regularized convex optimization problems. Our proposed method approximates a variable metric proximal\u2013gradient step with extrapolation by performing a prefixed number of primal\u2013dual iterates, while adjusting the steplength parameter through an appropriate backtracking procedure. Choosing a prefixed number of inner iterations allows the algorithm to keep the computational cost per iteration low. We prove the convergence of the iterates sequence towards a solution of the problem, under a relaxed monotonicity assumption on the scaling matrices and a shrinking condition on the extrapolation parameters. 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