Circular Motion without Physics

Geometry Level 5

Let $$ABC$$ be a triangle. Let $$I$$ be its incenter. Let $$L, M, N$$ be the circumcenters of triangles $$BIC, AIC, AIB$$, respectively. What is the sum of the powers of $$L, M, N$$ with respect to the circumcircle of $$\triangle ABC$$?

Note:The power of point P to circle $$\omega$$ with radius $$r$$ and center $$O$$ is $$OP^2 - r^2$$.

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