
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
A point $P(-\sqrt{3}, 2)$ is inside the circle $x^{2}+y^{2}=12$. $AB$ is a chord passing through point $P$ with a slope angle of $α$.

(1) Write out the maximum and minimum values of $|AB|$;

(2) When $α=120^{\circ}$, find $|AB|$;

(3) Find the trajectory equation of the midpoint of the chord passing through point $P$.

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