
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 the polar equation of a circle is $\rho^{2}-4 \sqrt{2}\rho\cos \left( \theta- \frac{\pi}{4} \right)+6=0$, find:

$(1)$ The standard equation and parametric equation of the circle;

$(2)$ The maximum and minimum values of $xy$ for all points $(x,y)$ on the circle.

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