
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 rectangular coordinate system $xOy$, the parametric equations of curve $C$ are $\left\{{\begin{array}{l}{x=2+3\cos\alpha}\\{y=3\sin\alpha}\end{array}}\right.$ ($\alpha$ is the parameter). Taking the coordinate origin $O$ as the pole and the non-negative half-axis of the $x$-axis as the polar axis, the polar coordinate equation of the line $l$ is $2\rho \cos \theta -\rho \sin \theta -1=0$.  
$(1)$ Find the general equation of curve $C$ and the rectangular coordinate equation of line $l$;  
$(2)$ If line $l$ intersects curve $C$ at points $A$ and $B$, and point $P(0,-1)$, find the value of $\frac{1}{{|{PA}|}}+\frac{1}{{|{PB}|}}$.

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