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update a value entry following some rule

i have a vector $(x_1, \dots,x_n)$ where $x_i$ belongs to some set $E$ how can i assign randomly a letter $a,b,c$ to each vector than update it following some rule these vector ?

Example $E={ x_1, x_2,x_3 }$

$(x_1, x_2,x_3)=a$




$(x_1,x_1,x_1)=whatevre$ etc...

i have a rule that if $(x_1, x_2,x_3)=a$ and $(x_2,x_3,x_1)=a$ then $(x_3,x_1,x_2)$ must be equal to $a$