Fix a large natural number $n$. Let $a_1,a_2,\dots,a_n$ be $n$ real numbers satisfying
$$ (1+a_i-a_j)(1+a_j-a_k)(1+a_k-a_i)=(1+a_j-a_i)(1+a_i-a_k)(1+a_k-a_j) $$ for any $1\le i,j,k\le n$.
How to find all solutions to the above system of equations? I can see that, obviously, a constant sequence is a solution. But I don't know if there are other solutions.