
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 parametric equation of circle $C$ is given by
$$
\begin{cases}
x = 2 + 2\cos \theta \\
y = 2\sin \theta
\end{cases}
\quad (\theta \text{ is the parameter}).
$$
The equation of line $l_1$ is $x + 1 = 0$. With $O$ as the pole and the non-negative half-axis of $x$ as the polar axis, establish a polar coordinate system.

(1) Find the polar equation of circle $C$ and line $l_1$.
(2) If the polar equation of line $l_2$ is $\theta = \frac{\pi}{3} (\rho \in \mathbb{R})$, and line $l_2$ intersects circle $C$ at points $O$ and $P$ (where $O$ is the origin), with line $l_2$ intersecting line $l_1$ at point $Q$, find the length of $|PQ|$.

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