
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Positive real numbers \( x, y, z \) satisfy
\[
\left\{
\begin{array}{l}
\frac{2}{5} \leqslant z \leqslant \min \{x, y\}, \\
xz \geqslant \frac{4}{15}, \\
yz \geqslant \frac{1}{5}.
\end{array}
\right.
\]

Find the maximum value of \( \frac{1}{x} + \frac{2}{y} + \frac{3}{z} \).

Present the answer in LaTex format: \boxed{Your answer}
user