Geometry Level 5

If $$a$$ is a real constant and $$A,B$$ and $$C$$ are variable angles. They are related to the constraint $$\sqrt{a^{2}-13}\tan{A} + a\tan{B} + \sqrt {a^{2}+13}\tan{C} = 6a$$ . Then find minimum positive value of $$\tan^{2}{A} + \tan^{2}{B} + \tan^{2}{C}$$.

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