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A Bayesian calibration, using experimental data from 2.76 A TeV Pb-Pb collisions at the LHC, of a novel hybrid model is presented in which the usual pre-hydrodynamic and viscous relativistic fluid dynamic (vRFD) stages are replaced by a viscous anisotropic hydrodynamic (VAH) core that smoothly interpolates between the initial expansion-dominated, approximately boost-invariant longitudinally free-streaming and the subsequent collision-dominated (3+1)- dimensional standard vRFD stages. This model yields meaningful constraints for the temperature-dependent specific shear and bulk viscosities, (η=s)(T) and (ζ=s)(T), for temperatures up to about 700MeV (i.e. over twice the range that could be explored with earlier models). With its best-fit model parameters the calibrated VAH model makes highly successful predictions for additional pT-dependent observables for which high-quality experimental data are available that were not used for the model calibration. [ABSTRACT FROM AUTHOR]