Traingle with inverse angles

Geometry Level 3

$\tan^{-1} \left ( \frac 1 2 \right), \tan^{-1} \left ( \frac 12 + k \right), \tan^{-1} \left ( \frac 1 2 + 2k \right )$

Find the sum of all value(s) of $$k$$ for which there is a triangle whose angles have measures given above.

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