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Geometry Level 4

Let $$\alpha, \beta, \gamma$$ be positive numbers satisfying $$\alpha + \beta + \gamma = \frac \pi 2$$. And denote $$A,B,C$$ that satisfies

$\large\ \begin{cases} A=\tan { \alpha } \tan { \beta } +5 \\ B = \tan { \beta } \tan { \gamma } + 5 \\ C = \tan { \gamma } \tan { \alpha } + 5 \end{cases}$.

Find maximum value of $$\sqrt { A } +\sqrt { B } +\sqrt { C }$$.