
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
Given in the Cartesian coordinate system, the parametric equation for curve C_1 is 
$$
\begin{cases}
x = \sqrt{3} + 3\cos\phi \\
y = -1 + 3\sin\phi
\end{cases}
$$
($\phi$ is the parameter). Establishing a polar coordinate system with the origin as the pole and the positive x-axis as the polar axis, the polar equation for curve C_2 is $\rho = 2\cos\theta$.

(I) Find the polar equation for curve C_1 and the Cartesian equation for curve C_2;
(II) If the line $\theta = \frac{\pi}{6}$ ($\rho \in \mathbb{R}$) intersects with curve C_1 at points P and Q, find the length of $|PQ|$.

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