
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 equation of the line $l_{1}$ is $y=k(x-\sqrt{3})$, and the parameter equation of the line $l_{2}$ is $\left\{\begin{array}{l}{x=-\sqrt{3}+t}\\{y=-\frac{1}{k}t}\end{array}\right.$ (where $t$ is the parameter). Let $P$ be the intersection point of $l_{1}$ and $l_{2}$. As $k$ varies, the locus of $P$ is the curve $C$.  
$(1)$ Find the general equation of $C$.  
$(2)$ The line $l$ passing through $Q(0,2)$ intersects $C$ at points $A$ and $B$. Find the range of values of $\frac{1}{{|QA|}}+\frac{1}{{|QB|}}$.

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