\[\int_{\pi }^{2\pi} \lfloor 2 \sin x \rfloor \, dx\]

If the value of the integral above is of the form \(\dfrac{A\pi }{B}\), where \(A\) and \(B\) are coprime integers, find \(\left | A \right |+\left | B \right |\).

**Notations**:

\( \lfloor \cdot \rfloor \) denotes the floor function.

\( | \cdot | \) denotes the absolute value function.

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