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Property, Plant, and Equipment (Details) - USD ($)
$ in Millions
3 Months Ended 9 Months Ended
May 27, 2023
May 28, 2022
May 27, 2023
May 28, 2022
Aug. 27, 2022
Property, Plant and Equipment [Line Items]          
Property, plant, and equipment, gross $ 518.5   $ 518.5   $ 455.3
Less: Accumulated depreciation 198.5   198.5   179.1
Property, plant, and equipment, net 320.0   320.0   276.2
Depreciation 7.6 $ 6.3 20.9 $ 17.0  
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 232.1   232.1   171.0
Machinery and equipment          
Property, Plant and Equipment [Line Items]          
Property, plant, and equipment, gross 156.3   156.3   142.6
Software          
Property, Plant and Equipment [Line Items]          
Property, plant, and equipment, gross 49.8   49.8   43.8
Transportation          
Property, Plant and Equipment [Line Items]          
Property, plant, and equipment, gross 7.1   7.1   6.5
Construction in progress          
Property, Plant and Equipment [Line Items]          
Property, plant, and equipment, gross $ 58.6   $ 58.6   $ 76.8