Let \(f(x)=x-\dfrac{1}{x}\).

If \((a_1,b_1)\cdots (a_n,b_n)\) are the solutions to the system of equations

\[\left\{\begin{array}{l}f(a)+f(b)=f(ab)\\ a=f(b)\end{array}\right.\]

then the value of \[\sum_{i=1}^n|a_i|+|b_i|\] can be expressed as \[\sqrt{a}+\sqrt{b}+\sqrt{c}\] for positive integers \(a,b,c\).

Find \(a+b+c\).

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