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

$(1)$ Write the standard equation of line $l$ and the Cartesian coordinate equation of curve $C$;  
$(2)$ Let $M(x,y)$ be any point on curve $C$, find the minimum value of $|x-y-4|$.

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