
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 plane, the line $l$ passes through the point $P(1,1)$, and its inclination angle $\alpha= \frac{\pi}{4}$. Taking the origin $O$ as the pole and the non-negative half-axis of the $x$-axis as the polar axis, a polar coordinate system is established. The polar equation of circle $C$ is given by $\rho=4\sin \theta$.

$(1)$ Find the Cartesian equation of circle $C$;

$(2)$ Suppose the line $l$ intersects circle $C$ 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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