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Goodwill and Intangible Assets - Rollforward (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
Goodwill [Roll Forward]          
Goodwill, beginning balance     $ 114.8 $ 3.5  
Additions to Goodwill     77.3 111.3  
Goodwill impairment charge $ (10.3) $ (30.7) (41.0) 0.0 $ (147.7)
Goodwill, ending balance 151.1   151.1 114.8 3.5
Science, Engineering & Technology          
Goodwill [Roll Forward]          
Goodwill, beginning balance     111.3 0.0  
Additions to Goodwill     0.0 111.3  
Goodwill impairment charge     0.0 0.0  
Goodwill, ending balance 111.3   111.3 111.3 0.0
Education          
Goodwill [Roll Forward]          
Goodwill, beginning balance     3.5 3.5  
Additions to Goodwill     36.3 0.0  
Goodwill impairment charge     0.0 0.0  
Goodwill, ending balance 39.8   39.8 3.5 $ 3.5
Outsourcing & Consulting          
Goodwill [Roll Forward]          
Goodwill, beginning balance 10.3   0.0    
Additions to Goodwill     41.0    
Goodwill impairment charge     (41.0)    
Goodwill, ending balance $ 0.0 $ 10.3 $ 0.0 $ 0.0