
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 $ρ^2 - 4\sqrt{2}ρ\cos\left(θ- \frac{π}{4}\right) +6=0$.

$(1)$ Convert the polar equation into a Cartesian equation and express it using an appropriate parametric equation.

$(2)$ If point $P(x,y)$ lies on the circle, find the maximum and minimum values of $x+y$.

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