The maximum value of \[\sin 2x +2 \sin x \] can be expressed as \({\dfrac{ a \sqrt{b}}{c}}\), where \(a\) and \(c\) are co-prime positiive integers and \(b\) is a square-free integer. Find the value of \({\dfrac{1}{a}+\dfrac{1}{c}+\dfrac{1}{bc}} \).

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