Solving $x=|tau(x)|^{1/beta-1}(beta-1)^{1/beta}frac{pi/beta}{sin(pi/beta)}-|tau(x)|^{-1}+o(|tau(x)|^{-1})$ for $tau(x)$

How do you solve the following equation for $tau(x)$?
begin{equation}
x=|tau(x)|^{1/beta-1}(beta-1)^{1/beta}frac{pi/beta}{sin(pi/beta)}-|tau(x)|^{-1}+o(|tau(x)|^{-1})
end{equation}

The author of the paper (https://arxiv.org/pdf/1607.08794.pdf, page 13) states that
begin{equation}
tau(x)=x^{-1}-b(beta)x^{-beta/(beta-1)}+o(x^{-1}),quad b(beta):=(beta-1)^{1/(beta-1)}left(frac{pi/beta}{sin(pi/beta)}right)^{beta/(beta-1)}
end{equation}

is the solution, but I wonder how to get there.
For the function $tau(x)$ holds $lim_{xrightarrow 0}=-infty, lim_{xrightarrow infty}=(beta-1)^{-1}$ and $tau(1)=0$, $beta>1$.
Another user has helped me to understand that
begin{equation}
tau(x)sim -b(beta)x^{-beta/(beta-1)} text{ for } xrightarrow0.
end{equation}

(Approximating $tau(x)$ from $x=|tau(x)|^{1/beta-1}(beta-1)^{1/beta}frac{pi/beta}{sin(pi/beta)}-|tau(x)|^{-1}+o(|tau(x)|^{-1})$), but I’m not sure if that helps/how to apply that for this problem.
Any help is appreciated.