Let f be a polynomial. We say that a complex number p is a double attractor if there exists a polynomial h(x) so that
f(x)-f(p)=h(x)(x-p)^{2} for all x \in \mathbb{R}. Now, consider the polynomial
$$f(x)=12 x^{5}-15 x^{4}-40 x^{3}+540 x^{2}-2160 x+1
and suppose that it's double attractors are $a_{1}, a_{2}, \ldots, a_{n}$. If the sum $\sum_{i=1}^{n}\left|a_{i}\right|$ can be written as $\sqrt{a}+\sqrt{b}$, where $a, b$ are positive integers, find $a+b$.