Let $u = \epsilon e^{\frac{1}{2} \eta}$ where
\begin{equation\*}
\left(
 \begin{array}{c}
 \epsilon \\
 \eta \\
 \end{array}
\right)
\sim
N\left(
\left(
 \begin{array}{c}
 0 \\
 0 \\
 \end{array}
\right),
\left(
 \begin{array}{cc}
 1 & \rho \sigma \\
 \rho \sigma & \sigma^2 \\
 \end{array}
\right)
\right),
\quad
-1 < \rho < 1.
\end{equation\*}
What is $\mathbb{E}[u \eta]$?

Thank you.