
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, with the origin as the pole and the positive half-axis of the $x$-axis as the polar axis, a polar coordinate system is established. It is known that the polar equation of curve $C$ is $\rho^2-4\sqrt{3}\rho\sin\left(\theta+ \frac{\pi}{3}\right)=8$. The curve $C$ intersects with the line $l_1: y= \sqrt{3}x$ at points $A$ and $B$, and the parametric equation of line $l_2$ is $\begin{cases}x=- \frac{1}{2}x+ \sqrt{3}t \\ y=- \frac{ \sqrt{3}}{2}- \frac{1}{2}t\end{cases}$ (where $t$ is the parameter), and $l_1$ intersects with $l_2$ at point $P$. Find the values of $|PA|$ and $|PB|$.

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