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Algorithms for solving agreement problems can be classified in two categories: (1) those relying on failure detectors that we call \emph{FD-based}, and (2) those that rely on a Group Membership Service that we call ``GMS-based''. The paper discusses the advantages and limitations of these two approaches, and proposes an extension to the GMS-approach that combines the advantages of both approaches, without their drawbacks. This extension leads us to distinguish between ``time-triggered'' suspicions of processes and ``space-triggered'' exclusions.