An Interesting equation

\[\left\{\lfloor x\rfloor + \frac{x}{\sqrt{3}}\right\} - \left\{\frac{\sqrt{3}}{x}\right\}=0; 1\leq x\leq 2\] If twice the sum of all possible real values of \(x\) satisfying the equation above is of the form \(\sqrt{a} + \sqrt{ab}\), where \(a,b\) are integers, find the value of \(a+b\) **Notations:** - \(\lfloor\cdot\rfloor\) denoted the floor function - \(\{\cdot\}\) denotes the fractional part function