
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 Cartesian coordinate system $xOy$, given circle $O: x^{2}+y^{2}=4$ and point $P(-1,1)$, the line $l$ passing through point $P$ intersects circle $O$ at points $A$ and $B$.  
$(1)$ If $|AB|=2\sqrt{3}$, find the equation of line $l$;  
$(2)$ Let the midpoint of chord $AB$ be $M$, find the equation of the trajectory of point $M$.

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