
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 equation of curve $C_1$ is $y=k|x|+2$. With the origin as the pole and the positive $x$-axis as the polar axis, a polar coordinate system is established, and the polar equation of curve $C_2$ is $p^2+2p\cos \theta-3=0$.

$(1)$ Find the Cartesian equation of $C_2$;

$(2)$ If $C_1$ and $C_2$ have exactly three points in common, find the equation of $C_1$.

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