
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 polar coordinate system, given the curve $C: \rho = 2\cos \theta$, the line $l: \left\{ \begin{array}{l} x = \sqrt{3}t \\ y = -1 + t \end{array} \right.$ (where $t$ is a parameter), and the line $l$ intersects the curve $C$ at points $A$ and $B$.  
$(1)$ Find the rectangular coordinate equation of curve $C$ and the general equation of line $l$.  
$(2)$ Given the polar coordinates of point $P$ as $({1, \frac{3\pi}{2}})$, find the value of $\left(|PA|+1\right)\left(|PB|+1\right)$.

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