The polar coordinates of points $A,$ $B,$ and $C$ are $(2,70^\circ),$ $(2 \sqrt{3}, 40^\circ),$ and $(3,10^\circ),$ respectively.  Compute the area of triangle $ABC.$The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.

Let's think step by step and output the final answer within \boxed{}.
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