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Revenue Recognition - Narrative (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
Revenue from Contract with Customer [Abstract]          
Percentage of revenues derived from route servicing fees 88.50% 91.00% 88.40% 91.20%  
Current assets related to deferred commissions $ 16.1   $ 16.1   $ 15.2
Noncurrent assets related to deferred commissions 69.1   69.1   $ 65.1
Amortization expense related to deferred commissions $ 4.2 $ 3.9 $ 12.3 $ 11.5