Admissions Role of AdS/CFT correspondence in the context of integrability

I have a masters degree. I studied general relativity and quantum field theory. I was interested in applying to PhD programs for AdS/CFT. I was wondering how integrability fits in the context of AdS/CFT. As I understand, the AdS/CFT correspondence postulates a duality between gravity theories and CFT's. If one theory has a strong coupling, the other has weak coupling. AdS/CFT would then be used to study the strongly coupled theory by analyzing the weakly coupled dual theory instead. However, with integrability, (I think) one can directly analyze the strongly coupled theory. Where would AdS/CFT correspondence fit in this situation?
 

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