
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 \( x \) and \( y \) for which the expression

\[ 
\frac{x y}{2}+\frac{18}{x y} 
\]

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

\[ 
\frac{y}{2}+\frac{x}{3} 
\]

is also minimized.

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