
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 in the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is $\begin{cases} x=\frac{\sqrt{2}}{2}t \\ y=\frac{\sqrt{2}}{2}t+4\sqrt{2} \end{cases}$ (where $t$ is the parameter), and establishing a polar coordinate system with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of curve $C$ is $\rho =2\cos (\theta +\frac{\pi }{4})$.

$(1)$ Determine the positional relationship between line $l$ and curve $C$;

$(2)$ Let $M$ be any point on curve $C$, find the range of values for $x+y$.

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