
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
In the Cartesian coordinate system $xOy$, the equation of line $l$ is $x-y+4=0$, and the parametric equation of curve $C$ is $\begin{cases} & x=\sqrt{3}\cos \alpha \\ & y=\sin \alpha \\ \end{cases}$ ($\alpha$ is the parameter).

$(1)$ Given in the polar coordinate system (with the same unit length as the Cartesian coordinate system $xOy$, and the origin $O$ as the pole, and the positive half-axis of $x$ as the polar axis), the polar coordinates of point $P$ are $(2\sqrt{2} ,\frac{3\pi }{4} )$, determine the positional relationship between point $P$ and line $l$;

$(2)$ Let point $Q$ be a moving point on curve $C$, find the minimum distance from it to line $l$.

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