
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 $\left\{\begin{array}{l}{x=\sqrt{2}\cos\theta+2}\\{y=\sqrt{2}\sin\theta}\end{array}\right.$ ($\theta$ is the parameter). Line $l$ passes through the origin and makes an angle $\alpha$ with the positive $x$-axis. Establish a polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis.<br/>$(1)$ Find the polar coordinate equation of curve $C$ and line $l$;<br/>$(2)$ Given that curve $C$ intersects line $l$ at points $A$ and $B$, and $|OA|+|OB|=3$, find the Cartesian equation of line $l$.

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