
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 rectangular coordinate system, a pole coordinate system is established with the origin as the pole and the positive semi-axis of the $x$-axis as the polar axis. Given the curve $C$: ${p}^{2}=\frac{12}{2+{\mathrm{cos}}^{}θ}$ and the line $l$: $2p\mathrm{cos}\left(θ-\frac{π}{6}\right)=\sqrt{3}$.

1. Write the rectangular coordinate equations for the line $l$ and the curve $C$.
2. Let points $A$ and $B$ be the two intersection points of line $l$ and curve $C$. Find the value of $|AB|$.

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