For every $a\in\Bbb{R}$ find the number of solutions for $e^{-|x|}(x-1)^3=a$

I'm struggling with the following problem:

For every $a\in\Bbb{R}$ find the number of solutions (where $x\in\Bbb{R}$) for $e^{-|x|}(x-1)^3=a$

After some transformations we get $(x-1)^3-ae^{|x|}=0$. Now I had an idea of finding critical points of the derivative of this function and examining its extremums, but the derivative is still going to have $e$ at some point so it leads to a similair problem. So, how to examine all $a$'s at once? Are the derivatives the way to go?

$$y(x)=e^{-|x|}(x-1)^3$$
The solution to equation in question is intersection point of graph of $y$ and graph of $y=a$. Now that greatly simplifies the problem.
Notice also that as $x\to \pm \infty$, we have $y\to 0$ as $y=O(e^{-|x|})$ (exponential functions dominate polynomials for large $x$).
$y=0$ if and only if $x=1$ and is $y>0$ for $x>1$, $y<0$ for $x<1$.
Can you imagine sketch of $y$ now? Then only thing you need to find is critical points of $y$, that is, where the maxima/minima occurs. Then you can easily classify number of solutions according to value of $a$ (and for this use of derivatives is probably best method).