  We perform Floquet spectroscopy in a GaAs double quantum dot system coupled
to a high-impedance superconducting resonator. By applying microwave induced
consecutive passages under a double resonance condition, we observe novel
Landau-Zener-St\"uckelberg-Majorana interference patterns that stem from a
cavity-assisted interference pattern modified by the depletion of the ground
state. Our experimental results reveal the stationary state behavior of a
strongly driven two-level system, and are consistent with the simulations based
on our theoretical model. This study provides an excellent platform for
investigating the dynamics of Floquet states in the presence of strong driving.
