
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 \(xOy\), establishing a polar coordinate system with the origin as the pole and the positive \(x\)-axis as the polar axis, the polar equation of curve \(C_1\) is given by \(\rho^2 - 4\rho\cos\theta + 3 = 0\), where \(\theta\in[0, 2\pi]\).
1. Find the Cartesian equation of curve \(C_1\).
2. The parametric equations of curve \(C_2\) are:
   \[
   \begin{cases} 
   x = t\cos \frac{\pi}{6} \\ 
   y = t\sin \frac{\pi}{6}
   \end{cases}
   \]
   (where \(t\) is the parameter). Find the polar coordinates of the intersection points of \(C_1\) and \(C_2\).

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