I'm interested in solving the differential equation

$$
a\frac{d y}{dx} = -\frac{1}{y(x)} e^{-\frac{1}{y(x)}}
$$
where $a>0$. 

$Mathematica$ can solve this but it gives the answer in terms of InverseFunctions of exponential integrals. I'm only interested in the asymptotic behavior of $y(x)$ as $x \to \infty$ but I'm not sure how to proceed with this sort of a problem. 

Edit: The suggestion was to transform to $z(x) = \frac{1}{y(x)}$. In terms of this new variable, the problem becomes
$$
a \frac{dz}{dx} = z(x)^3 e^{-z(x)}
$$
As far as initial conditions are concerned, let's say $z(0) = 1$. Even in these new variables, I don't really know how to proceed.