Let $a,b,c$ be real variables.
Let $z$ be a complex number and $g(z) = exp(z)(z-a)(z-b)(z-c)$.
Let $f(z)$ be the functional inverse of $g(z)$ such that $f(g(z)) = g(f(z)) = z$. Now $f(z)$ must have singularities where $g'(z) = 0$ hence since this equation has at most 3 distinct complex solutions and $g(z)$ is an entire function ; $f(z)$ has at most 3 distinct singularities.
Lets call those singularities $A,B,C$.
Now I wonder about Puiseux series
What are the Puiseux series at the points $A,B,C$ ?
I note that although not written in the link , a Puiseux series power series part is suppose to have nonzero radius of convergence.
Depending (for instance) on how the radius of convergence is for the power series part , there are at most 4 possible Puiseux series for point $A$ because there are 3 singularities. ( those being valid everywhere , valid close too from A to B , valid close too from A to B and C , valid close too from A to C )
In fact the amount of Puiseux series up to translation with $m$ singularities is at most $2^m - 1$. And that makes $7$ in this case.
So what are the Puiseux series for the inverse of $f(z)$ expanded at the singularities?