HMMT November 2021 U18

Let $x, y, z$ be real numbers satisfying

$$\frac{1}{x}+y+z=x+\frac{1}{y}+z=x+y+\frac{1}{z}=3 .$$

The sum of all possible values of $x+y+z$ can be written as $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n)=1$. Find $100 m+n$.