Algebra Level 5

Consider the following system of inequalities:

$\begin{cases} (c-1)x^2+2cx+c+4\leq 0\\ cx^2+2(c+1)x+(c+1)\geq 0 \end{cases}$

The sum of all real values of $c$, such that the system has a unique solution, can be written as $\frac{a}{b}$, where $a$ and $b$ are coprime positive integers. What is the value of $a+b$?

Details and assumptions

$c$ can be negative.

The system has a unique solution if there is only 1 real value $x$ which is satisfied throughout.

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