  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.
