Extrema(s) of Trigonometric Functions!

Geometry Level 5

\[\large{\left( \sin(A) + \sin(B) + \sin(C) \right) \left( \dfrac1A + \dfrac1B + \dfrac1C \right)}\]

Let \(A,B,C\) be the angles (in radians) of a triangle. If the minimum value of the above expression can be expressed as:

\[\large{\dfrac{\sqrt{A^B}}{C \cdot \pi^D}}\]

such that \(A, B, C,D\) are positive integers, find the minimum value of \(A+B+C+D\)?

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