
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 $xOy$, a line $l$ passes through the point $P(2,-1)$ with an inclination angle of $45^{\circ}$. Taking the origin as the pole and the positive half-axis of $x$ as the initial line to establish the polar coordinate system, the polar equation of curve $C$ is $\rho\sin^2\theta=4\cos\theta$. The intersection points of line $l$ and curve $C$ are $A$ and $B$.

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

$(2)$ Find $|PA|\cdot|PB|$.

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