Googly Eyed Triangle

Geometry Level 4

Point \(D\) is on side \(BC\) of \(\triangle ABC.\) The incircle \(\omega _1\) of \(\triangle ABD\) is tangent to \(BC\) at \(X\) and to \(AD\) at \(Y.\) The incircle \(\omega _2\) of \(\triangle ADC\) is tangent to \(BC\) at \(Z\) and to \(AD\) at \(Y.\) If the radius of \(\omega _1\) is \(2,\) and the radius of \(\omega _2\) is \(1,\) find the value of \(XY^2 + YZ^2\) to the nearest tenth.

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