Real nonzero numbers $x, y, z$ satisfy $x + y + z = 0$. Furthermore, it is given that:
\[
A = \frac{x}{y} + \frac{y}{z} + \frac{z}{x} = \frac{x}{z} + \frac{z}{y} + \frac{y}{x} + 1
\]
Determine the value of $A$.

Let's think step by step and output the final answer within \boxed{}.
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