Can someone explain the meaning of “symmetry” and how is that equality assumed in the comment for the editorial of Div2E Tetrahedron

Can someone help please ??

By symmetry we mean that since there is an edge from every vertex to every other it does not matter which specific vertex we are at. So, for example if A_i denotes the number of ways we can end up at vertex A after i moves, this would be equal to B_i and C_i (due to this “symmetry”). This simplifies the recurrence as A_i = B_i = C_i.

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