
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 circle $M$: $(x+1)^{2}+y^{2}=1$, and circle $N$: $(x-1)^{2}+y^{2}=9$, a moving circle $P$ is externally tangent to circle $M$ and internally tangent to circle $N$. The trajectory of the center of circle $P$ is curve $C$.
$(1)$ Find the equation of $C$.
$(2)$ Let $l$ be a line tangent to both circle $P$ and circle $M$, and $l$ intersects curve $C$ at points $A$ and $B$. When the radius of circle $P$ is the longest, find $|AB|$.

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