A multilevel four-stroke engine where the thermalization stokes are generated by unitary collisions with bath particles is analyzed. Our model is solvable even when the engine operates far from thermal equilibrium and in the strong system-bath coupling. Necessary operation conditions for the heat machine to perform as an engine or a refrigerator are derived. We relate the work and efficiency of the device to local and non-local statistical properties of the baths (purity, mutual coincidence etc.). In particular, we relate the Clausius inequality to the symmetrized relative entropy of the baths (Jefferys divergence). Other Clausius-like inequalities are derived as well. Finally, in the ultra-hot regime we optimize the work of the multilevel engine and obtain simpler forms for the work and efficiency.