
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
In the Cartesian coordinate system, it is known that the focal points of the ellipse are located at $F_1(-2\sqrt{2}, 0)$ and $F_2(2\sqrt{2}, 0)$. The minimum distance from a point on ellipse $C$ to the right focal point is $3 - 2\sqrt{2}$.  
(1) Determine the equation of ellipse $C$.  
(2) If a line with a slope of -2 intersects the ellipse $C$ at points $E$ and $F$, find the equation of the trajectory of the midpoint $N$ of segment $EF$.  
(3) Suppose a line passing through the point $F_1(-2\sqrt{2}, 0)$ intersects the ellipse $C$ forming a chord $PQ$. Find the maximum area of triangle $\Delta PQO$, where $O$ is the origin of the coordinates.

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