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Numerous quarkonium(like) states lying near S-wave thresholds are observed experimentally. We propose a self-consistent approach to these near-threshold states compatible with unitarity and analyticity. The underlying coupled-channel system includes a bare pole and an arbitrary number of elastic and inelastic channels treated fully nonperturbatively. The resulting analytical parametrization is ideally suited for a combined analysis of the data available in various channels that is exemplified by an excellent overall description of the data for the charged Z(b)(10610) and Z(b)(10650) states.