Maximum Value

Geometry Level 3

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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