The chemotaxis network in Escherichia coli is remarkable for its sensitivity to small relative changes in the concentrations of multiple chemical signals over a broad range of ambient concentrations. Key to this sensitivity is an adaptation system that relies on methylation and demethylation (or deamidation) of specific modification sites of the chemoreceptors by the enzymes CheR and CheB, respectively. It was recently discovered that these enzymes can access five to seven receptors when tethered to a particular receptor. We show that these "assistance neighborhoods" are necessary for precise adaptation in a model for signaling by clusters of chemoreceptors. In agreement with experiment, model clusters composed of receptors of different types exhibit high sensitivity and precise adaptation over a wide range of chemical concentrations and the response of adapted clusters to addition/removal of attractant scales with free-energy change. We predict two limits of precise adaptation at large attractant concentrations: Either receptors reach full methylation and turn off, or receptors become saturated and cease to respond to attractant but retain their adapted activity.