Our talk introduces a one-reduce GMRES algorithm based on Gauss-Seidel (MGS) and Jacobi (CGS) iterations, together with a new AMG smoother. The correction matrix $T = (I + L)^{-1}$ for the projector $P = I - QTQ^T$ is approximated using a rank-1 perturbation of the identity, resulting in a low backward error. These ideas are applied to the AMG preconditioner. Inspired by Eirola and Nevanlinna (1989), the V-cycle pre-smoother first performs an $O(n^2)$ triangular solve and subsequent iterations apply Jacobi or the $O(n)$ product $(I - \gamma D^-1uv^T)D^{-1}r_k$ where $u = L_{k,1:k-1}$ and $v = e_k$, with shifts. The post-smoother updates a vector $x_{k+1}$ with one Gauss-Seidel or Jacobi iteration. The proposed approach is an efficient algebraic multigrid smoother whose convergence can be analysed with Neumann proxies and Gershgorin circles. Results from ill-conditioned Navier-Stokes pressure solvers exhibit a 3X decrease in compute time on GPUs. This iterative refinement approach is most effective when the $\kappa(D+L)$ is large, and convergence is accelerated by shifts.