Pensions and Other Benefit Plans - Plan with Projected Benefit Obligation in Excess of Plan Assets (Details) - USD ($) $ in Thousands	3 Months Ended		6 Months Ended		12 Months Ended
	Jul. 02, 2023	Jul. 03, 2022	Jul. 02, 2023	Jul. 03, 2022	Jan. 01, 2023	Jan. 02, 2022	Jan. 03, 2021
Change in Plan Assets
Plan assets at fair value - beginning of year			$ 19,000	$ 0	$ 0
Plan assets at fair value - end of year					19,000	$ 0
Amounts Recognized in the Company’s Balance Sheet consist of the following:
Accrued liabilities					(7,000)	(6,000)
Employee related obligations - non-current					(214,000)	(302,000)
Total recognized in the Combined Balance Sheets - end of year					(221,000)	(308,000)
Retirement Plans
Change in Benefit Obligation
Projected benefit obligation - beginning of year			235,000	303,000	303,000	347,000
Service cost	$ 5,000	$ 2,000	10,000	4,000	8,000	7,000	$ 6,000
Interest cost	$ 7,000	$ 1,000	10,000	2,000	4,000	2,000	3,000
Actuarial gain					(82,000)	(21,000)
Curtailments, settlements & restructuring					0	0
Benefits paid from plan					(8,000)	(9,000)
Effect of exchange rates					(19,000)	(23,000)
Other					29,000	0
Projected benefit obligation - end of year					235,000	303,000	347,000
Change in Plan Assets
Plan assets at fair value - beginning of year			19,000	0	0	0
Company contributions					9,000	9,000
Benefits paid from plan assets					(8,000)	(9,000)
Actual return on plan assets					(1,000)	0
Effect of exchange rates					(2,000)	0
Transfers					21,000	0
Plan assets at fair value - end of year					19,000	0	0
Funded status - end of year					(216,000)	(303,000)
Amounts Recognized in the Company’s Balance Sheet consist of the following:
Accrued liabilities					(7,000)	(6,000)
Employee related obligations - non-current					(209,000)	(297,000)
Total recognized in the Combined Balance Sheets - end of year					(216,000)	(303,000)
Amounts Recognized in Accumulated Other Comprehensive Income consist of the following:
Net actuarial (gain) loss					(15,000)	79,000
Prior service cost					4,000	2,000
Total before tax effects					(11,000)	81,000
Accumulated Benefit Obligations - end of year					204,000	262,000
Retirement Plans | Defined Benefit Plan, Unfunded Plan
Change in Benefit Obligation
Other					25,000
Other Benefit Plans
Change in Benefit Obligation
Projected benefit obligation - beginning of year			$ 5,000	5,000	5,000	5,000
Service cost					0	0	0
Interest cost					0	1,000	0
Actuarial gain					0	0
Curtailments, settlements & restructuring					0	0
Benefits paid from plan					0	0
Effect of exchange rates					0	(1,000)
Other					0	0
Projected benefit obligation - end of year					5,000	5,000	5,000
Change in Plan Assets
Plan assets at fair value - beginning of year				$ 0	0	0
Company contributions					0	0
Benefits paid from plan assets					0	0
Actual return on plan assets					0	0
Effect of exchange rates					0	0
Transfers						0
Plan assets at fair value - end of year						0	$ 0
Funded status - end of year					0	0
Amounts Recognized in the Company’s Balance Sheet consist of the following:
Accrued liabilities					0	0
Employee related obligations - non-current					(5,000)	(5,000)
Total recognized in the Combined Balance Sheets - end of year					(5,000)	(5,000)
Amounts Recognized in Accumulated Other Comprehensive Income consist of the following:
Net actuarial (gain) loss					(4,000)	(4,000)
Prior service cost					0	0
Total before tax effects					(4,000)	(4,000)
Accumulated Benefit Obligations - end of year					$ 3,000	$ 3,000