If \( x,y,\) and \(z \) are real numbers that satisfy the following system of equations:

\[\begin{align} x^2 +6y & = -14 \\ y^2 + 12z &= -63 \\ z^2 - 4x &= 28, \end{align}\]

what is the value of \(\lvert x + y + z \rvert\)?

This problem is posed by Danny H.

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