
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 parametric equations of curve $C$ are $\left\{\begin{array}{l}x=\sqrt{2}({\sin\alpha-\cos\alpha})\\ y=\frac{{\sqrt{2}}}{2}({\sin\alpha+\cos\alpha})\end{array}\right.$ ($\alpha$ is the parameter). Establish a polar coordinate system with the origin $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis. The polar coordinate equation of the line $l$ is $\rho \cos \theta +2\rho \sin \theta +m=0$.  
$(1)$ Find the Cartesian equations of curve $C$ and line $l$.  
$(2)$ If the maximum distance from a point on curve $C$ to line $l$ is $\frac{{4\sqrt{10}}}{5}$, find the value of the real number $m$.

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