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Including phonon-assisted inelastic process in thermoelectric devices is able to enhance the performance of nonequilibrium work extraction. In this work, we demonstrate that inelastic phonon-thermoelectric devices have a fertile functionality diagram, where particle current and phononic heat currents are coupled and fueled by chemical potential difference. Such devices can simultaneously perform multiple tasks, e.g., heat engines, refrigerators, and heat pumps. Guided by the entropy production, we mainly study the efficiencies and coefficients of performance of multitask quantum thermal machines, where the roles of the inelastic scattering process and multiple biases in multiterminal setups are emphasized. Specifically, in a three-terminal double-quantum-dot setup with a tunable gate, we show that it efficiently performs two useful tasks due to the phonon-assisted inelastic process. Moreover, the cooperation between the longitudinal and transverse thermoelectric effects in the three-terminal thermoelectric systems leads to markedly improved performance of the thermal machines. While for the four-terminal four-quantum-dot thermoelectric setup, we find that additional thermodynamic affinity furnishes the system with both enriched functionality and enhanced efficiency. Our work provides insights into optimizing phonon-thermoelectric devices.