
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 rectangular coordinate system $(xOy)$, the parametric equations of line $l$ are $\begin{cases} x=\frac{1}{2}t \\ y=3+\frac{\sqrt{3}}{2}t \end{cases} (t \text{ is the parameter})$, and in the polar coordinate system with the pole as $O$ and the positive semi-axis of $x$ as the polar axis, the polar equation of curve $C$ is $\rho=2\sin\theta$.
(1) Find the ordinary equation of line $l$ and the rectangular coordinate equation of curve $C$;
(2) Let point $P$ be the intersection of line $l$ and the $y$-axis, and points $A, B$ be the intersections of line $l$ and curve $C$. Find the value of $|PA| \cdot |PB|$.

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