Forecasting the length of conditionally Gaussian vector

Question

Assume that $S(t)=\sqrt{X(t)^2+Y(t)^2}$, where $X$ and $Y$ are conditionally Gaussian, so $X_{t_n+h}|X_{t_1},\dots,X_{t_n}$ and $Y_{t_n+h}|Y_{t_1},\dots,Y_{t_n}$ have Gaussian distributions. I'm able to forecast $X$ and $Y$, i.e. evaluate $E[X_{t_n+h}|X_{t_1},\dots,X_{t_n}]$ and $\mathbb E[Y_{t_n+h}|Y_{t_1},\dots,Y_{t_n}]$, with $h>0$. Of course $E[S_{t_n+h}|X_{t_1},\dots,X_{t_n},Y_{t_1},\dots,Y_{t_n}]\neq \sqrt{E[X_{t_n+h}|X_{t_1},\dots,X_{t_n}]^2+E[Y_{t_n+h}|Y_{t_1},\dots,Y_{t_n}]^2}$.
Question: How can I obtain forecast $S$ given forecasts of $X$ and $Y$?