Minimizing Tangents!

Geometry Level 4

\[\sqrt{a^2-4}\tan{A} + a\tan{B} + \sqrt{a^2+4}\tan{C} = 6a \]

If \(a\) is a real constant and \(A,B,C\) are variable angles such that the equation above is fulfilled, find the least value of \(\tan^{2}A + \tan^{2}B +\tan^{2}C \).

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