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Goodwill and Intangible Assets - Narrative (Details) - USD ($)
$ in Millions
3 Months Ended 6 Months Ended
Feb. 29, 2024
Feb. 28, 2023
Feb. 29, 2024
Feb. 28, 2023
Goodwill and Intangible Assets Disclosure [Abstract]        
Amortization of intangible assets $ 10.0 $ 9.3 $ 19.9 $ 22.9