Average Sorting

You are given a sequence $A_1, A_2, \ldots, A_N$. You may perform the following operation any number of times: select any two adjacent elements $A_i$ and $A_{i + 1}$ and replace one of them with $\frac{A_i + A_{i + 1}}{2}$ (a real number, i.e. without rounding). You may also select each pair of adjacent elements and each element to replace in multiple operations, i.e. any number of times. Is it p