Let \(x = \sin\left(\sin^{-1} \left(\frac{3}{5}\right) + \tan^{-1} (2)\right)\). \(x\) can be written in the form \(\frac{a\sqrt{b}}{c}\), where \(a, b\) and \(c\) are positive integers, \(a\) and \(c\) are coprime and \(b\) is not divisible by the square of any prime. What is the value of \(a+b+c\)?

**Details and assumptions**

\(\sin^{-1}\) and \(\tan^{-1}\) represent the inverse of the \(\sin\) and \(\tan\) function, and not the reciprocal.

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