
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 $C$: $(x-1)^2 + y^2 = 16$, point $F(-1, 0)$, and $M$ is a moving point on circle $C$. The perpendicular bisector of line segment $MF$ intersects with line segment $MC$ at point $P$.

(I) Find the trajectory equation $E$ of point $P$;

(II) Line $l$ passes through point $(-1, 0)$, intersecting trajectory $E$ at points $A$ and $B$, with $O$ being the coordinate origin. Find the maximum area of triangle $OAB$.

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