
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 line $l$ passes through the point $P(0,\frac{1}{2})$ and has an inclination angle of $150^{\circ}$. With $O$ as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of circle $C$ is $\rho^2 + 2\rho \cos \theta = 0$ (where $\theta$ is the parameter, $\rho > 0$).  
$(1)$ Write the parametric equation of line $l$ and the Cartesian equation of circle $C$;  
$(2)$ Suppose line $l$ intersects circle $C$ at points $A$ and $B$, find the value of $|PA|\cdot|PB|$.

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