
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 rectangular coordinate system $(xOy)$, the equation of line $l$ is given by $\begin{cases} x=\sqrt{3}+\frac{\sqrt{2}}{2}t \\ y=2-\frac{\sqrt{2}}{2}t. \end{cases}$ (where $t$ is a parameter). Taking the origin $O$ as the pole and the $Ox$ axis as the polar axis, establish a polar coordinate system with the same unit length. The equation of curve $C$ is given by $\rho =2\sqrt{3}\cos \theta$.

(1) Find the rectangular coordinate equation of curve $C$.
(2) Suppose curve $C$ intersects with line $l$ at points $A$ and $B$, and let $P(\sqrt{3},2)$. Find $|PA|+|PB|$ and $|AB|$.

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