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Geometry Level 4

$\dfrac{1}{90} \displaystyle \sum^{90}_{n=1} 2n \sin(2n^\circ)$

If the value of the above expression is in the form $$\cot(k^\circ)$$ where $$0<k<180$$, find $$k$$.

Clarification: All angles are measured in degrees.

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