
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 line $l$ passes through the point $P(3, \sqrt{5})$ with a slope angle of $\frac{3}{4} \pi$. With the origin $O$ as the pole and the positive half of the $x$-axis as the polar axis, establish a polar coordinate system. The polar coordinate equation of the circle $C$ is $\rho = 2\sqrt{5} \sin \theta$.

(1) Write a parametric equation for the line $l$ and the rectangular coordinate equation for circle $C$.
(2) Let circle $C$ intersect with line $l$ 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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