
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 $xOy$, the parametric equation of line $l$ passing through point $P(2,1)$ is $\begin{cases}x=2+ \sqrt{3}t \\ y=1+t\end{cases}$ (where $t$ is the parameter). With the origin as the pole and the positive $x$-axis as the polar axis, the polar coordinate equation of curve $C$ is $\rho \sin^2\theta=2\cos\theta$. It is known that line $l$ intersects curve $C$ at points $A$ and $B$.
$(1)$ Find the Cartesian coordinate equation of curve $C$;
$(2)$ Find the value of $|PA|\cdot|PB|$.

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