
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 $(xOy)$, a line $l$ passes through point $P(\frac{1}{2}, 1)$ with an angle of inclination $\alpha = \frac{\pi}{6}$. In the polar coordinate system, which has the same unit length and origin $O$ as the cartesian system and the positive semi-axis of $x$ as the polar axis, the polar equation of circle $C$ is $\rho = 2\sqrt{2} \cos(\theta - \frac{\pi}{4})$.

1. Find the parametric equation of line $l$ and the cartesian equation of circle $C$.
2. If line $l$ intersects circle $C$ at points $A$ and $B$, find $|PA| \cdot |PB|$.

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