Question 121
Determine all positive $x, y, z$, that satisfy the following system of inequalities:
$$
\begin{cases} (x+1)(y+1) \le (z+1)^2, \\ \left(\frac{1}{x}+1\right)\left(\frac{1}{y}+1\right) \le \left(\frac{1}{z}+1\right)^2. \end{cases}
$$
$$
\begin{cases} (x+1)(y+1) \le (z+1)^2, \\ \left(\frac{1}{x}+1\right)\left(\frac{1}{y}+1\right) \le \left(\frac{1}{z}+1\right)^2. \end{cases}
$$