
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 the curve \(C_{1}\) is \(y=k|x|+2\). Taking the origin as the pole and the positive \(x\)-axis as the polar axis, the polar equation of the curve \(C_{2}\) is \(\rho^{2}+2\rho \cos \theta-3=0\).
\((1)\) Find the Cartesian coordinate equation of \(C_{2}\);
\((2)\) If \(C_{1}\) intersects with \(C_{2}\) at exactly three distinct points, determine the equation of \(C_{1}\).

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