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Transactions with Related Parties (Details)
$ in Thousands
9 Months Ended
Oct. 02, 2022
USD ($)
number_of_restaurants
Oct. 03, 2021
USD ($)
Jan. 02, 2022
USD ($)
Related Party Transaction      
Number of Restaurants | number_of_restaurants 7,080    
TimWen | Franchise Rental Expense      
Related Party Transaction      
Related Party Transaction, Expenses from Transactions with Related Party $ 15,148 $ 13,994  
TimWen | General and administrative | Management Fee Income      
Related Party Transaction      
Related Party Transaction, Other Income from Transactions with Related Party 166 165  
Yellow Cab | Royalty, Advertising Fund, Lease and Other Income      
Related Party Transaction      
Related Party Transaction, Other Income from Transactions with Related Party 9,942 $ 7,016  
Yellow Cab | Accounts and notes receivable, net and Advertising funds restricted assets | Royalty, Advertising Fund, Lease and Other Income      
Related Party Transaction      
Accounts Receivable, Related Parties, Current $ 987   $ 974
Yellow Cab | Franchised Units      
Related Party Transaction      
Number of Restaurants | number_of_restaurants 84