
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
Given \( z_{1}=x+\sqrt{5}+y i \) and \( z_{2}=x-\sqrt{5}+y i \), where \( x, y \in \mathbf{R} \) and \(\left|z_{1}\right|+\left|z_{2}\right|=6 \), find the product of the maximum and minimum values of \( f(x, y)=|2 x-3 y-12| \).

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