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Strong nonlinear materials have been sought after for decades for applications in telecommunications, sensing, and quantum optics. Gallium-doped zinc oxide is a II-VI transparent conducting oxide that shows promising nonlinearities similar to indium tin oxide and aluminum-doped zinc oxide for the telecommunications band. Here we explore its nonlinearities in the epsilon near zero (ENZ) region and show n2,eff values on the order of 4.5x10-3 cm2GW-1 for IR pumping on 200-300 nm thin films. Measuring nonlinear changes in transmission and reflection with a white light source probe in the near-IR while exciting in the near-IR provides data in both time and wavelength. Three films varying in thickness, optical loss, and ENZ crossover wavelength are numerically modeled and compared to experimental data showing agreement for both dispersion and temporal relaxation. In addition, we discuss optimal excitation and probing wavelengths occur around ENZ for thick films but are red-shifted for thin films where our model provides an additional degree of freedom to explore. Obtaining accurate nonlinear measurements is a difficult and time-consuming task where our method in this paper provides experimental and modeled data to the community for an ENZ material of interest.