
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 $xOy$, line $l$ passes through 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, a polar coordinate system is established. The polar equation of circle $C$ is $\rho^2 + 2\rho\cos\theta = 0$ (where $\theta$ is a parameter, $\rho > 0$).  
(1) Write the parametric equation of line $l$ and the Cartesian coordinate 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}
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