
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 plane Cartesian coordinate system $xOy$, the parametric equations of curve $C_{1}$ are $\left\{\begin{array}{l}{x=1+\frac{\sqrt{3}t}{2},}\\{y=\frac{t}{2}}\end{array}\right.$ (where $t$ is the parameter). Using the origin $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis, the polar coordinate equation of curve $C_{2}$ is $\rho ^{2}-4\rho \sin \theta =2$.  
$(1)$ Find the Cartesian equation of curve $C_{1}$ and the rectangular coordinate equation of curve $C_{2}$.  
$(2)$ Given $P(1,0)$, curve $C_{1}$ intersects curve $C_{2}$ at points $A$ and $B$. Find $|PA|\cdot |PB|$.

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