Implicit Modified Euler

Consider the implicit modified Euler numerical integration technique. The difference between this and its explicit counterpart is that the "dot" and "double-dot" terms have the "k" subscript instead of the "k-1" subscript.

$\large{x_k = x_{k-1} + \dot{x}_k \, \Delta t + \ddot{x}_k \, (\Delta t)^2 }$

Suppose we use this to discretely approximate the following continuous-time signal:

$\large{x(t) = e^{\alpha t}}$

In the above equation, $\alpha$ is a complex number. Define "stability" as the following being true for all processing steps:

$\large{\Big| \frac{x_k}{x_{k-1}} \Big| < 1}$

For $\Delta t = 1$ , how much area in the complex $\alpha$ plane is NOT included in the stability region?

Note: Even if the numerical results are "stable" (non-divergent), that doesn't necessarily mean that they are correct
Bonus: What do you notice about the stability properties of the implicit method, relative to the explicit method?