
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 parameter equation of the line $l$ is $\left\{{\begin{array}{l}{x=3-\frac{{\sqrt{3}}}{2}t,}\\{y=\sqrt{3}-\frac{1}{2}t}\end{array}}\right.$ (where $t$ is the parameter). Establish a polar coordinate system with the origin $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis. The polar coordinate equation of the curve $C$ is $ρ=2sin({θ+\frac{π}{6}})$.  
$(1)$ Find the general equation of the line $l$ and the rectangular coordinate equation of the curve $C$;  
$(2)$ If the polar coordinates of point $P$ are $({2\sqrt{3},\frac{π}{6}})$, the line $l$ intersects the curve $C$ at points $A$ and $B$. Find the value of $\frac{1}{{|{PA}|}}+\frac{1}{{|{PB}|}}$.

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