Question 1
Let $x$, $y$, $z$ be positive real numbers satisfying the following conditions:
$$
\begin{gathered}
\frac{1}{\sqrt{2}} \leq z \lt \frac{1}{2} \min \{x\sqrt{2},\ y\sqrt{3}\}, \\
x + z\sqrt{3} \geq \sqrt{6}, \\
y\sqrt{3} + z\sqrt{10} \geq 2\sqrt{5}.
\end{gathered}
$$
Find the greatest value of the expression:
$$
P(x, y, z) = \frac{1}{x^2} + \frac{2}{y^2} + \frac{3}{z^2}
$$
$$
\begin{gathered}
\frac{1}{\sqrt{2}} \leq z \lt \frac{1}{2} \min \{x\sqrt{2},\ y\sqrt{3}\}, \\
x + z\sqrt{3} \geq \sqrt{6}, \\
y\sqrt{3} + z\sqrt{10} \geq 2\sqrt{5}.
\end{gathered}
$$
Find the greatest value of the expression:
$$
P(x, y, z) = \frac{1}{x^2} + \frac{2}{y^2} + \frac{3}{z^2}
$$