Given a list $\overline{x} = [x_1, x_2, \dots, x_n]$ of integers where each $x_i$ is of size $n^{O(1)}$, determine whether there exist $1\leq i, j\leq n$ such that $i\neq j$, $j-i$ is even, and $x_{(i+j)/2} = \frac{1}{2}(x_i+x_j)$.