
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 the line $l$ are $\left\{{\begin{array}{l}{x=\frac{{\sqrt{2}}}{2}⋅t}\\{y=2+\frac{{\sqrt{2}}}{2}⋅t}\end{array}}\right.$ (where $t$ is the parameter). Using the origin as the pole and the positive x-axis as the polar axis, the polar coordinate equation of curve $C$ is $\rho =r\left(r \gt 0\right)$.  
$(1)$ Find the polar coordinate equation of the line $l$ and the Cartesian equation of curve $C$.  
$(2)$ Let point $M(0,2)$, the line $l$ intersects curve $C$ at points $A$ and $B$. If $\overrightarrow{MA}+3\overrightarrow{MB}=0$, find the Cartesian equation of curve $C$ in this case.

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