# Three Magnetically Coupled Coils

The frame shown on the left is made of material with magnetic permeability $$\mu >> \mu_0$$. The magnetic field in each segment of the frame is uniform and parallel to that segment. Coil circle has $$2 N$$ turns, coil square has $$N$$ turns, and coil triangle has $$N / 2$$ turns, where $$N >> 1$$ is an even integer. The dimensions of each segment of the frame are listed in the table below, where $$L >> \sqrt{A}$$:

 Segment Length Cross-sectional Area $$IM$$ $$4 L$$ $$A$$ $$JN$$ $$4 L$$ $$2 A$$ $$KO$$ $$4 L$$ $$A$$ $$IJ$$ $$2.5 L$$ $$A$$ $$JK$$ $$2.5 L$$ $$A$$ $$MN$$ $$2.5 L$$ $$A$$ $$NO$$ $$2.5 L$$ $$A$$

The coils are connected as shown on the right to form a single equivalent inductance. The equivalent inductance can be expressed as:

$$\frac{X}{Y}\frac{\mu N^2 A}{L}$$

where $$X$$ and $$Y$$ are coprime positive integers. Determine $$X + Y$$.

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