\[ \int_0^\pi \sqrt{ (\cos x + \cos 2x + \cos 3x)^2 + (\sin x + \sin2x + \sin3x)^2} \, dx \]

If the value of the integral above equals \[ \dfrac \pi A + B\sqrt C , \] where \(A,B\) and \(C\) positive integers with \(C\) square-free, submit your answer as \(A+B+C\).

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