OMCC 2022 Problem 2

Ana, Beto, Carlos, Diana, Elena and Fabian are in a circle, located in that order. Ana, Beto, Carlos, Diana, Elena and Fabian each have a piece of paper, where are written the real numbers $a,b,c,d,e,f$ respectively. At the end of each minute, all the people simultaneously replace the number on their paper by the sum of three numbers; the number that was at the beginning of the minute on his paper and on the papers of his two neighbors. At the end of the minute $2022, 2022$ replacements have been made and each person have in his paper it´s initial number. Find all the posible values of $abc+def$ . $\textbf{Note:}$ If at the beginning of the minute $N$ Ana, Beto, Carlos have the numbers $x,y,z$ , respectively, then at the end of the minute $N$ , Beto is going to have the number $x+y+z$.