Systems of Differential Equations · Hard problem

Let $A=\begin{pmatrix}-1&-3\\3&-1\end{pmatrix}$ and $\mathbf x'=A\mathbf x$ with $\mathbf x(0)=(1,0)$; the point spirals into the origin, rotating at a constant rate of $3$ radians per unit time while its distance from the origin decays exactly like $e^{-t}$. Find the total angle (in radians) through which the point has turned by the time its distance from the origin has shrunk by a factor of $e$.