Loci From Foci

Geometry Level 4

Two points $$F_1,F_2$$ and a point $$P$$ satisfy that $$|F_1P-F_2P|=2$$ and $$F_1F_2=4$$. Let $$I$$ be the incenter of $$\triangle F_1PF_2$$. As $$P$$ varies, $$I$$ traces out two disjoint loci. What is the sum of the lengths of these two loci? Round to the nearest thousandths.

$$\text{Details and Assumptions}$$

The incircle is drawn inaccurately on purpose.

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