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Revenue Recognition - Narrative (Details) - USD ($)
$ in Millions
3 Months Ended 9 Months Ended
May 28, 2022
May 29, 2021
May 28, 2022
May 29, 2021
Aug. 28, 2021
Revenue From Contract With Customer [Abstract]          
Percentage of revenues derived from route servicing fees 91.00% 90.90% 91.20% 91.00%  
Current assets related to deferred commissions $ 14.9   $ 14.9   $ 14.2
Noncurrent assets related to deferred commissions 64.1   64.1   $ 60.6
Amortization expense related to deferred commissions $ 3.9 $ 3.6 $ 11.5 $ 10.8