
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, the curve $C_1$ is defined by $\left\{ \begin{matrix} x=3+3\cos \alpha \\ y=2\sin \alpha \end{matrix} \right.$ (with $\alpha$ as the parameter) undergoes a scaling transformation $\begin{cases} x'= \frac{x}{3} \\ y'= \frac{y}{2} \end{cases}$ to become curve $C_2$. Establish a polar coordinate system with the origin as the pole and the positive x-axis as the polar axis.

(1) Find the polar equation of $C_2$;  
(2) Suppose the polar equation of curve $C_3$ is $\rho\sin \left( \frac{\pi }{6}-\theta \right)=1$, and curve $C_3$ intersects with curve $C_2$ at points $P$ and $Q$, find the value of $|PQ|$.

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