
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 two circles ${F_1: (x+1)^2 + y^2 = 9}$ and ${F_2: (x-1)^2 + y^2 = 1}$, a moving circle $ABC$ is tangent to the inside of $F_1$ and to the outside of $F_2$. $O$ is the origin of the coordinate system.

(I) Find the equation of the trajectory $PBQ$ of the center of the moving circle $ABC$.

(II) A line ${l: y = kx - 2}$ intersects the curve $PBQ$ at two points $A$ and $B$. Find the maximum area of $\triangle OAB$ and the equation of the line $l$ when the area is maximum.

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