
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 a circle with the polar equation $\rho^2 - 4\sqrt{2}\rho\cos\left(\theta - \frac{\pi}{4}\right) + 6 = 0$,  
find:  
(1) the standard form of the equation of the circle and a parametric equation;  
(2) the maximum and minimum value of $xy$ for all points $(x, y)$ on the circle.

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