We study a model of a thermoelectric nanojunction driven by vibrationally-assisted tunneling. We apply the reaction coordinate formalism to derive a master equation governing its thermoelectric performance beyond the weak electron-vibrational coupling limit. Employing full counting statistics we calculate the current flow, thermopower, associated noise, and efficiency without resorting to the weak vibrational coupling approximation. We demonstrate intricacies of the power-efficiency-precision trade-off at strong coupling, showing that the three cannot be maximised simultaneously in our model. Finally, we emphasise the importance of capturing non-additivity when considering strong coupling and multiple environments, demonstrating that an additive treatment of the environments can violate the upper bound on thermoelectric efficiency imposed by Carnot.
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