
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 given by $\begin{cases} x=\frac{\sqrt{2}}{2}t \\ y=3+\frac{\sqrt{2}}{2}t \end{cases}$ ($t$ is the parameter), and the polar coordinate equation of curve $C$ is $\rho =4\sin \theta -2\cos \theta$ in the polar coordinate system with $O$ as the pole and the positive $x$-axis as the polar axis.

(1) Find the Cartesian equation of line $l$ and curve $C$.
(2) If the point of intersection of line $l$ and the $y$-axis is $P$, and the points of intersection of line $l$ and curve $C$ are $A$ and $B$, find the value of $|PA|\cdot |PB|$.

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