
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]


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In the plane rectangular coordinate system $xOy$, with $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. The polar coordinate equation of curve $C$ is $\rho ^{2}\cos ^{2}\theta +3\rho ^{2}\sin ^{2}\theta =12$. Point $P$ has polar coordinates $\left(2,\pi \right)$, and the line $l$ with an inclination angle of $\alpha$ passes through point $P$.  
$(1)$ Write down the rectangular coordinate equation of curve $C$ and the parametric equation of line $l$.  
$(2)$ Suppose line $l$ intersects curve $C$ at points $A$ and $B$, find the range of $\frac{1}{{|PA|}}+\frac{1}{{|PB|}}$.

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