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Nature of Business, Basis of Presentation and Summary of Significant Accounting Policies - Narrative (Details)
$ in Thousands
 12 Months Ended
Oct. 03, 2025
USD ($)
segment
country
Sep. 27, 2024
USD ($)
Sep. 29, 2023
USD ($)
tranche
Sep. 30, 2022
USD ($)
Basis of Presentation and Summary of Significant Accounting Policies [Line Items]        
Number of reportable segments | segment 2      
Pension plan adjustments from spin-off   $ 5,100    
Foreign currency translation adjustments, spin-off   26,300    
Cash and cash equivalents $ 639,095 672,483 $ 1,927,088  
Inventory valuation reserves 10,000 19,300    
Depreciation 302,000 276,200 267,900  
Asset write-downs 8,946 18,186 29,865  
Equity method investment, aggregate cost 70,600 84,000    
Impairment of an equity investment 19,500      
Equity securities without readily determinable fair value, amount 59,600 35,400    
Proceeds from sale of equity investments 0 101,198 633,179  
Supplier finance program, obligation $ 4,724 $ 2,576    
Supplier Finance Program, Obligation, Statement of Financial Position [Extensible Enumeration] Accounts payable Accounts payable    
Payments related to tax withholding for share-based compensation $ 33,800 $ 21,400 31,300  
Equity Method Investments        
Basis of Presentation and Summary of Significant Accounting Policies [Line Items]        
Pre-tax gain on sale of equity investment     (377,100)  
Gain on sale of this equity investment     278,700  
Property and Equipment and Long-term Borrowings        
Basis of Presentation and Summary of Significant Accounting Policies [Line Items]        
Future rental obligations $ 25,900 13,300 4,900  
CARES Act        
Basis of Presentation and Summary of Significant Accounting Policies [Line Items]        
Payment of deferred social security taxes       $ 47,600
San Antonia Spurs        
Basis of Presentation and Summary of Significant Accounting Policies [Line Items]        
Pre-tax gain on sale of equity investment   (25,100) (1,100)  
Proceeds from sale of equity investments   101,200 98,200  
Loss on sale of equity investment, net of tax   19,600 $ (2,200)  
AIM Services Co., Ltd | AIM Services Co., Ltd        
Basis of Presentation and Summary of Significant Accounting Policies [Line Items]        
Equity method investment, ownership interest, percent     50.00%  
Proceeds from sale of equity method investments     $ 535,000  
Building and Building Improvements | Minimum        
Basis of Presentation and Summary of Significant Accounting Policies [Line Items]        
Property, plant and equipment, estimated useful lives 10 years      
Building and Building Improvements | Maximum        
Basis of Presentation and Summary of Significant Accounting Policies [Line Items]        
Property, plant and equipment, estimated useful lives 40 years      
Service Equipment and Fixtures | Minimum        
Basis of Presentation and Summary of Significant Accounting Policies [Line Items]        
Property, plant and equipment, estimated useful lives 3 years      
Service Equipment and Fixtures | Maximum        
Basis of Presentation and Summary of Significant Accounting Policies [Line Items]        
Property, plant and equipment, estimated useful lives 20 years      
Captive        
Basis of Presentation and Summary of Significant Accounting Policies [Line Items]        
Cash and cash equivalents $ 133,500 94,700    
Investment at cost $ 0 42,300    
FSS United States        
Basis of Presentation and Summary of Significant Accounting Policies [Line Items]        
Inventory adjustment   18,200    
Gain on investments     $ 36,300  
Number of service sites | tranche     1  
FSS United States | Cost of services provided (exclusive of depreciation and amortization)        
Basis of Presentation and Summary of Significant Accounting Policies [Line Items]        
Asset write-downs     $ 19,000  
FSS United States | Right of Use Assets | Cost of services provided (exclusive of depreciation and amortization)        
Basis of Presentation and Summary of Significant Accounting Policies [Line Items]        
Asset write-downs     8,600  
FSS United States | Leasehold improvements | Cost of services provided (exclusive of depreciation and amortization)        
Basis of Presentation and Summary of Significant Accounting Policies [Line Items]        
Asset write-downs     10,400  
Foreign Countries Outside United States        
Basis of Presentation and Summary of Significant Accounting Policies [Line Items]        
Number of countries in which entity operates | country 15      
ARGENTINA        
Basis of Presentation and Summary of Significant Accounting Policies [Line Items]        
Foreign currency transaction loss $ 5,700 $ 5,400 $ 10,400