
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 Cartesian coordinate system $xOy$, the parametric equations of curve $C$ are $\left\{{\begin{array}{l}{x=\frac{1}{{\cos\alpha}},}\\{y=\frac{{\sqrt{3}\sin\alpha}}{{\cos\alpha}},}\end{array}}\right.$ ($\alpha$ is the parameter, $\alpha≠kπ+\frac{π}{2}$), with the 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 $ρ\cos({θ+\frac{π}{3}})=1$.  
$(1)$ Find the Cartesian equation of curve $C$ and the rectangular coordinate equation of line $l$;  
$(2)$ Given the point $P(2,0)$, if the line $l$ intersects 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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