
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 $xOy$, line $l$ passes through point $P(2, \sqrt{3})$ with an inclination angle of $\frac{3\pi}{4}$. In the polar coordinate system (using the same unit length as the Cartesian coordinate system $xOy$, with the origin as the pole and the positive x-axis as the polar axis), the equation of circle $C$ is $\rho=2\sqrt{3}\sin\theta$.  
(1) Find the parametric equation of line $l$ and the Cartesian coordinate equation of circle $C$;  
(2) Suppose line $l$ intersects circle $C$ at points $A$ and $B$, find $|PA|+|PB|$.

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