
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
Given points $A(-2,0)$, $B(2,0)$, and a moving point $P$ on curve $C$ satisfying $\overrightarrow{AP} \cdot \overrightarrow{BP} = -3$.

$(I)$ Find the equation of curve $C$;

$(II)$ If a line $l$ passing through the fixed point $M(0,-2)$ has common points with curve $C$, find the range of the slope $k$ of line $l$;

$(III)$ If a moving point $Q(x,y)$ is on curve $C$, find the range of $u= \frac{y+2}{x-1}$.

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