Functional equation with certain properties

Question

Let $a,b \in \Bbb{R}, b>0$. Find all the differentiable functions $f:\Bbb{R} \rightarrow \Bbb{R}$, for which $\lim_{x\to -\infty} f(x) $= a and which verify : $$ f'(x)=b(f(x)-a)^{2},$$ for every $x \in \Bbb{R}$
[Attempt] Suppose there exist $x_{0}$ such that $f(x_{0})\ne a $ and because f is continuous there exist an interval $I$ such that $f(x)\ne a, $ for every $x \in I$, then: $$ \int \left(\frac{-1}{f(x)-a}\right)' dx =\int \frac{f'(x)}{(f(x)-a)^2}dx= \int b \ dx \implies xb+c = \frac{-1}{f(x)-a} $$ so $ \dfrac{-1}{xb+c} +a =f(x)$ but from here I don't know how to get to a contradiction.