My second favourite problem ever

Algebra Level 3

If \(x,y,z\) are real numbers that satisfy the equation \( (x+y)^2 = 16, (y+z)^2 =36, (z+x)^2 = 81 \) and \(x+y+z>3 \). What is the number of possible values of \(x+y+z\)?

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