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Equity-Based Compensation - Stock Option Activity (Details) - Stock options - $ / shares
3 Months Ended 9 Months Ended 12 Months Ended
Oct. 02, 2022
Oct. 03, 2021
Oct. 02, 2022
Oct. 03, 2021
Jan. 02, 2022
Share-based Compensation Arrangement by Share-based Payment Award, Options, Outstanding [Roll Forward]          
Options Outstanding, Beginning balance     322,793    
Options Outstanding, Granted 0 322,793 0 322,793  
Options Outstanding, Ending balance 322,793   322,793   322,793
Options Outstanding, Vested and exercisable 161,397   161,397    
Options Outstanding, Expected to vest 322,793   322,793    
Share-based Compensation Arrangement by Share-based Payment Award, Options, Outstanding, Weighted Average Exercise Price [Abstract]          
Weighted-Average Exercise Price per Option, Beginning balance (in dollars per share)     $ 11.35    
Weighted-Average Exercise Price per Option, Ending balance (in dollars per share) $ 11.35   11.35   $ 11.35
Weighted-Average Exercise Price per Option, Vested and exercisable (in dollars per share) 11.35   11.35    
Weighted-Average Exercise Price per Option, Expected to vest (in dollars per share) $ 11.35   $ 11.35    
Share-based Compensation Arrangement by Share-based Payment Award, Options, Additional Disclosures [Abstract]          
Weighted-Average Remaining Contractual Life (years)     8 years 6 months 14 days   9 years 3 months 14 days
Weighted-Average Remaining Contractual Life (years), Vested and exercisable     8 years 6 months 14 days    
Weighted-Average Remaining Contractual Life (years), Expected to vest     8 years 6 months 14 days