The extent of nucleosynthesis in models of type I X-ray bursts and the associated impact on the energy released in these explosive events are sensitive to nuclear masses and reaction rates around the $^{64}$Ge waiting point. Using the well known mass of $^{64}$Ge, the recently measured $^{65}$As mass, and large-scale shell model calculations, we have determined new thermonuclear rates of the $^{64}$Ge($p$,$gamma$)$^{65}$As and $^{65}$As($p$,$gamma$)$^{66}$Se reactions with reliable uncertainties. The new reaction rates differ significantly from previously published rates. Using the new data we analyze the impact of the new rates and the remaining nuclear physics uncertainties on the $^{64}$Ge waiting point in a number of representative one-zone X-ray burst models. We find that in contrast to previous work, when all relevant uncertainties are considered, a strong $^{64}$Ge $rp$-process waiting point cannot be ruled out. The nuclear physics uncertainties strongly affect X-ray burst model predictions of the synthesis of $^{64}$Zn, the synthesis of nuclei beyond $A=64$, energy generation, and burst light curve. We also identify key nuclear uncertainties that need to be addressed to determine the role of the $^{64}$Ge waiting point in X-ray bursts. These include the remaining uncertainty in the $^{65}$As mass, the uncertainty of the $^{66}$Se mass, and the remaining uncertainty in the $^{65}$As($p$,$gamma$)$^{66}$Se reaction rate, which mainly originates from uncertain resonance energies.
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