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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