
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
Find all positive \( x, y, z \) for which the expression

$$
\frac{x^{2} y z}{324}+\frac{144 y}{x z}+\frac{9}{4 x y^{2}}
$$

takes the minimum value, and among all such \( x, y, z \), the expression

$$
\frac{z}{16 y}+\frac{x}{9}
$$

is also minimized.

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