Let $x_1, x_2, \dots, x_n$ be positive real numbers that are not all equal, with arithmetic mean $A$ and geometric mean $G$. If we add a new positive value $x_{n+1}$ to the dataset, under what condition will the new geometric mean $G_{\text{new}}$ be strictly greater than $G$, while the new arithmetic mean $A_{\text{new}}$ is strictly less than $A$?