Need an explanation of a particular expression transformation

Please, I need an explanation of the one transformation. I have the equation set and its solution.

$$\begin{cases} \frac{x}{y} + \frac{y}{z} + \frac{z}{x} = 3\\\\ \frac{y}{x} + \frac{z}{y} + \frac{x}{z} = 3\\\\ \ x + y + z = 3 \end{cases}$$

In the solution three new variables were introduced:

$$u = \frac{x}{y}; v = \frac{y}{z}; w = \frac{z}{x}$$

And then using this new vars equation set became this:

$$\begin{cases} \ u + v + w = 3\\\\ \frac{1}{u} + \frac{1}{v} + \frac{1}{w} = 3\\\\\\ \ uvw = 1 \end{cases}$$

I can't understand how the $x + y + z = 3$ has become the $uvw = 1$. Can someone explain what have been done here?

My appreciation.

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$$uvw = \frac{x}{y}\frac{y}{z} \frac{z}{x}=\frac{xyz}{yzx}=1$$