
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 $\left\{{\begin{array}{l}{x=-1+2\cos\alpha}\\{y=2+2\sin\alpha}\end{array}}\right.$ ($\alpha$ is the parameter). Taking the coordinate origin $O$ as the pole and the non-negative half-axis of the $x$-axis as the polar axis, the polar coordinate equation of the line $l$ is $\rho \cos \theta -2\rho \sin \theta +4=0$.  
$(1)$ Find the general equation of curve $C$ and the rectangular coordinate equation of line $l$;  
$(2)$ Given $P(-4,0)$, let line $l$ and curve $C$ intersect at points $A$ and $B$, with the midpoint of line segment $AB$ being $Q$, find the value of $|PQ|$.

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