# Law of Cosines

Geometry Level 5

Suppose that the angles of $$\triangle ABC$$ satisfy $$\cos(3A)+\cos(3B)+\cos(3C)=1.$$ Two sides of the triangle have lengths 10 and 13. There is a positive integer $$m$$ so that the maximum possible length for the remaining side of $$\triangle ABC$$ is $$\sqrt{m}.$$ Find $$m.$$

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