
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 parametric equations of curve $C$ are $\left\{\begin{array}{l}{x=2\cos t,}\\{y=\sin t}\end{array}\right.$ (where $t$ is the parameter). Taking the origin $O$ of the plane rectangular coordinate system as the pole and the positive half-axis of the $x$-axis as the polar axis, establish a polar coordinate system with the same unit of length. Let the polar equation of the line $l$ be $ρ\cos(θ+\frac{π}{3})=-\frac{\sqrt{3}}{2}$. <br/>$(1)$ Find the Cartesian equation of curve $C$ and the rectangular coordinate equation of line $l$; <br/>$(2)$ If line $l$ intersects curve $C$ at points $A$ and $B$, find the length of line segment $AB$.

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