
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 the line $l$ are: $\left\{{\begin{array}{l}{x=-1+t\cos\alpha}\\{y=-3+t\sin\alpha}\end{array}}\right.$ ($t$ is the parameter), with the coordinate origin as the pole and the non-negative half-axis of the $x$-axis as the polar axis, the polar coordinate equation of curve $C$ is $\rho =4\cos \theta$.  
$(1)$ When $α=\frac{π}{3}$, find the general equation of line $l$ and the rectangular coordinate equation of curve $C$.  
$(2)$ Line $l$ intersects curve $C$ at points $A$ and $B$, if $|AB|=2$, find the value of $\sin 2\alpha$.

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