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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 \(\sqrt { A } +\sqrt { B } +\sqrt { C }\).

Give your answer to 3 decimal places.

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