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Income Taxes - Narrative (Details) - USD ($)
$ in Thousands
 12 Months Ended
Sep. 29, 2023
Sep. 30, 2022
Oct. 01, 2021
Tax Credit Carryforward [Line Items]      
Current (benefit) provision for income taxes     $ 16,700
Tax benefit to the deferred (benefit) provision for income taxes related to a corresponding decrease in deferred tax liabilities     13,800
Net tax expense to the (benefit) provision for income taxes related to unrecognized tax benefits     2,900
Current income tax assets $ 10,198 $ 10,842  
Accrued taxes 25,000 2,600  
Proceeds from income tax refunds     93,600
Net tax expense to provision (benefit) for income taxes 76,700 (4,200)  
Gains and Losses from restructuring of certain foreign subsidiaries 21,700    
Income tax expense (benefit) 177,614 61,461 (40,633)
Net operating loss carried back to Pre-TCJA years     50,300
Valuation allowances 78,194 83,827 97,472
Decrease in income taxes receivable     1,800
Foreign tax credit carryforwards 74,400    
Tax credit carry forward, general business 800    
Deferred tax payable 583,832 470,606  
Operating loss carryforwards 85,500    
Undistributed foreign earnings 455,900 347,200  
Tax liability on undistributed foreign earnings 27,300 20,400  
Gross unrecognized tax benefits 70,313 80,220 65,414
Unrecognized tax benefits that would impact the effective tax rate 39,900    
Adjustment in deferred tax liability 30,400    
Accrued for interest and penalties 11,400 9,700  
Income tax penalties and interest expense 1,700 3,100 2,000
AIM Services Co., Ltd      
Tax Credit Carryforward [Line Items]      
Gains and Losses from restructuring of certain foreign subsidiaries 98,400    
Foreign Tax Credit Carryforward      
Tax Credit Carryforward [Line Items]      
Tax credit carryforward, amount 9,600    
Valuation Allowance, Operating Loss Carryforwards      
Tax Credit Carryforward [Line Items]      
Valuation allowances 47,100    
Deferred Income Taxes and Other Noncurrent Liabilities      
Tax Credit Carryforward [Line Items]      
Deferred tax payable 610,500 501,400  
Other assets      
Tax Credit Carryforward [Line Items]      
Deferred tax assets 26,700 30,800  
Foreign Tax Authority | Foreign Tax Credit Carryforward      
Tax Credit Carryforward [Line Items]      
Tax credit carryforward, Interest 31,100    
Foreign Tax Authority | Canada Revenue Agency      
Tax Credit Carryforward [Line Items]      
Tax adjustments, settlements, and unusual provisions 4,000    
State and Local Jurisdiction      
Tax Credit Carryforward [Line Items]      
Valuation allowances 10,100    
CARES Act      
Tax Credit Carryforward [Line Items]      
Income tax expense (benefit)     12,000
Valuation allowances     36,500
Income taxes receivable     3,700
Foreign tax credit carryforwards     71,300
Tax credit carry forward, general business     11,000
FSS International      
Tax Credit Carryforward [Line Items]      
Deferred tax asset, increase (decrease), amount $ 3,800 $ 8,500 $ (22,000)