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Goodwill and Intangible Assets - Narrative (Details) - USD ($) $ in Millions	3 Months Ended		12 Months Ended
	Jan. 01, 2023	Oct. 02, 2022	Jan. 01, 2023	Jan. 02, 2022	Jan. 03, 2021
Finite-Lived Intangible Assets [Line Items]
Finite-lived intangible assets, amortization expense, next 12 months	$ 20.8		$ 20.8
Finite-lived intangible assets, amortization expense, year 2	20.5		20.5
Finite-lived intangible assets, amortization expense, year 3	18.5		18.5
Finite-lived intangible assets, amortization expense, year 4	17.8		17.8
Finite-lived intangible assets, amortization expense, year 5	17.2		17.2
Goodwill impairment charge	10.3	$ 30.7	41.0	$ 0.0	$ 147.7
Goodwill, net	151.1		151.1	114.8	3.5
Outsourcing & Consulting
Finite-Lived Intangible Assets [Line Items]
Goodwill impairment charge			41.0
Goodwill, net	$ 0.0	$ 10.3	0.0	0.0
Selling, General and Administrative Expenses
Finite-Lived Intangible Assets [Line Items]
Amortization expense			$ 19.4	$ 13.0	$ 6.8