Prove that (i) \(x_1x_2x_3 + 1 = 0\), (ii) \(x_4 = x_1\).

It is given that
\[\begin{equation*}
x_2 = \frac{1}{1 - x_1}, \quad x_3 = \frac{1}{1 - x_2}, \quad x_4 = \frac{1}{1 - x_3}, \quad\text{where $x_1 \ne 0$ or $1$.}
\end{equation*}\]

Prove that (i) \(x_1x_2x_3 + 1 = 0\), (ii) \(x_4 = x_1\).