Systems of Differential Equations · Superchallenge problem

Every solution of $x'=-x+4y,\ y'=-y$ tends to the origin, yet some solutions first move away from it. Find the largest possible value of $\dfrac{\lVert\mathbf x(t)\rVert}{\lVert\mathbf x(0)\rVert}$ over all $t\ge0$ and all nonzero starting points $\mathbf x(0)$ (Euclidean length).