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Property, Plant, and Equipment (Details) - USD ($)
$ in Millions
3 Months Ended 6 Months Ended
Feb. 25, 2023
Feb. 26, 2022
Feb. 25, 2023
Feb. 26, 2022
Aug. 27, 2022
Property, Plant and Equipment [Line Items]          
Property, plant, and equipment, gross $ 501.7   $ 501.7   $ 455.3
Less: Accumulated depreciation 191.3   191.3   179.1
Property, plant, and equipment, net 310.4   310.4   276.2
Depreciation 6.7 $ 5.5 13.3 $ 10.8  
Land          
Property, Plant and Equipment [Line Items]          
Property, plant, and equipment, gross 14.6   14.6   14.6
Buildings and building improvements          
Property, Plant and Equipment [Line Items]          
Property, plant, and equipment, gross 201.6   201.6   171.0
Machinery and equipment          
Property, Plant and Equipment [Line Items]          
Property, plant, and equipment, gross 150.8   150.8   142.6
Software          
Property, Plant and Equipment [Line Items]          
Property, plant, and equipment, gross 44.1   44.1   43.8
Transportation          
Property, Plant and Equipment [Line Items]          
Property, plant, and equipment, gross 6.7   6.7   6.5
Construction in progress          
Property, Plant and Equipment [Line Items]          
Property, plant, and equipment, gross $ 83.9   $ 83.9   $ 76.8