The Reciprocal Twist
Algebra
Find the value of $x+y+z$ if $x$, $y$, and $z$ are positive real numbers that satisfy the equations below:
$$
\begin{cases}
xyz = 1\\
x+y+z = \frac{1}{x}+\frac{1}{y}+\frac{1}{z}\\
x^2 + y^2 + z^2 = 3
\end{cases}$$