
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 that O is the coordinate origin, point M is a moving point inside the circle O: x²+y²=4, and point F(1,0) is fixed. A circle is inscribed inside the given circle, having FM as its diameter.
(1) Find the equation of the trajectory C of the moving point M;
(2) Let point A(4,0). A line l passes through point F(1,0) and intersects the trajectory C at points P and Q. Find the maximum difference in the areas of triangles OAP and OAQ.

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