
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 equations of the line $l$ are $\left\{\begin{array}{l}{x=t,}\\{y=\sqrt{3}t-2}\end{array}\right.$ (where $t$ is a parameter). Taking $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis, the polar coordinate equation of the circle $C$ is $\rho ^{2}-4\rho \cos \theta +3=0$.  
$(1)$ Find the general equation of the line $l$ and the Cartesian equation of the circle $C$;  
$(2)$ If the coordinates of point $P$ are $\left(0,-2\right)$, and the line $l$ intersects the circle $C$ at points $A$ and $B$, find the value of $|PA|+|PB|$.

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