Classical geometry 2

Geometry Level 4

In the figure to the right, the yellow circle is tangent to \( \triangle ABC \) at points \( D, E, F. \)

\( 2 \sqrt{14}, 4 \sqrt{3}, \) and \( \sqrt{42} \) are the lengths of the common tangents to the semicircles of diameters \((BD, CD), (CE, AE),\) and \((AF, BF),\) respectively.

What is the radius of the yellow circle?

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