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Perfect adaptation is a well-studied biochemical homeostatic behavior lying at the core of biochemical regulation. While the concepts of homeostasis and perfect adaptation are not new, their underlying mechanisms and associated biochemical regulation motifs are not yet fully understood. Insights from control theory unraveled the connections between perfect adaptation and integral control, a prevalent engineering control strategy. In particular, the recently introduced Antithetic Integral Controller (AIC) has been shown to successfully ensure perfect adaptation properties to the network it is connected to. The complementary structure of the two molecules the AIC relies upon allows for a versatile way to control biochemical networks, a property which gave rise to an important body of literature pertaining to mathematically elucidating its properties, generalizing its structure, and developing experimental methods for its implementation. The Antithetic Integral Rein Controller (AIRC), an extension of the AIC in which both controller molecules are used for control, holds many promises as it supposedly overcomes certain limitations of the AIC. We focus here on an AIRC structure with output inhibition that combines two AICs in a single structure. We demonstrate that rhis controller ensure structural stability and structural perfect adaptation properties for the controlled network under mild assumptions, meaning that this property is independent of the parameters of the network and the controller. The results are very general and valid for the class of unimolecular mass-action networks as well as more general networks, including cooperative and Michaelis-Menten networks. We also provide a systematic and accessible computational way for verifying whether a given network satisfies the conditions under which the structural property would hold.