
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, the origin coincides with the pole of the polar coordinate system, with the positive x-axis as the polar axis. The line $l$ passes through the point $P(-1, 1)$, and the inclination angle $\alpha$ of the line is $\frac{5\pi}{6}$. The polar equation of curve $C$ is $\rho = 4\sin\theta$.
(Ⅰ) Find the parametric equation of line $l$ and the Cartesian equation of curve $C$;
(Ⅱ) Suppose line $l$ intersects curve $C$ at points $A$ and $B$. Find the value of $\vec{PA} \cdot \vec{PB}$.

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