
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 line $l$ is given by $\left\{{\begin{array}{l}{x=-\frac{1}{2}t}\\{y=3+\frac{{\sqrt{3}}}{2}t}\end{array}}\right.$ (where $t$ is a parameter). Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis. The polar coordinate equation of the curve $C$ is $ρ=4sin(θ+\frac{π}{3})$.  
$(1)$ Find the general equation of the line $l$ and the rectangular coordinate equation of the curve $C$.  
$(2)$ Let the coordinates of point $M$ be $\left(0,3\right)$. The intersection points of the line $l$ and the curve $C$ are $A$ and $B$. Find the value of $|AB|$.

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