Given four independent, identically distributed Gaussian random variables with zero mean and unit variance $x_1$, $x_2$, $y_1$, $y_2$, consider

\begin{equation} u \equiv \max(x_1+C\, y_1, x_2+C \, y_2) - \max(x_1-C \, y_1, x_2-C \, y_2), \end{equation}

where $C$ is a real number.

Do you know how to compute the PDF of $u$, or a least its variance?