Nonlinear Systems & Linearization · Superchallenge problem

The oscillator $x'' + \mu\left(|x| - 1\right)x' + x = 0$ (a van der Pol oscillator with $|x|$ in place of $x^2$) has a limit cycle for every $\mu > 0$. For large $\mu$ it is a relaxation oscillation, and its period satisfies $T(\mu) = c\mu + o(\mu)$ as $\mu \to \infty$. Find $c$.