
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 that the pole of the polar coordinate system coincides with the origin of the Cartesian coordinate system, and the polar axis coincides with the positive half-axis of the $x$-axis, if the polar equation of line $l$ is $p\sin (\theta- \frac {\pi}{4})=2 \sqrt {2}$.  
$(1)$ Convert the polar equation of line $l$ into an equation in the Cartesian coordinate system;  
$(2)$ Given that $P$ is a point on the ellipse $C$: $ \frac {x^{2}}{3}+ \frac {y^{2}}{9}=1$, find the minimum distance from $P$ to line $l$.

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