  We combine the formalisms of Floquet theory and full counting statistics with
a Markovian embedding strategy to access the dynamics and thermodynamics of a
periodically driven thermal machine beyond the conventional Born-Markov
approximation. The working medium is a two-level system and we drive the
tunneling as well as the coupling to one bath with the same period. We identify
four different operating regimes of our machine which include a heat engine and
a refrigerator. As the coupling strength with one bath is increased, the
refrigerator regime disappears, the heat engine regime narrows and their
efficiency and coefficient of performance decrease. Furthermore, our model can
reproduce the setup of laser cooling of trapped ions in a specific parameter
limit.
