Problem source

The $n$ positive  numbers  $x_{1},x_{2},\dots,x_{n}$, where  $n\ge3$, satisfy
$$
x_{1}=1+\frac{1}{x_{2}}\, ,\ \ \ 
x_{2}=1+\frac{1}{x_{3}}\, , \ \ \
\dots\; ,
\ \ \ x_{n-1}=1+\frac{1}{x_{n}}\, ,
$$
and also
$$
\ x_{n}=1+\frac{1}{x_{1}}\, .
$$
Show that
\begin{questionparts}
\item  $x_{1},x_{2},\dots,x_{n}>1$,
\item ${\displaystyle x_{1}-x_{2}=-\frac{x_{2}-x_{3}}{x_{2}x_{3}}}$,
\item $x_{1}=x_{2}=\cdots=x_{n}$.
\end{questionparts}
Hence find the value of $x_1$.