
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 $F_1$: $(x+1)^2+y^2=9$, and circle $F_2$: $(x-1)^2+y^2=1$, the moving circle $P$ is internally tangent to circle $F_1$ and externally tangent to circle $F_2$. Let $O$ be the origin of the coordinate system.  
(Ⅰ) Find the equation of the trajectory $C$ of the center $P$.  
(Ⅱ) The line $l: y=kx-2$ intersects curve $C$ at points $A$ and $B$. Find the maximum area of $\triangle OAB$ and the equation of line $l$ when the maximum area is achieved.

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