
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, the coordinates of the two foci of an ellipse are $F_{1}(-2\sqrt{2},0)$ and $F_{2}(2\sqrt{2},0)$. The minimum distance from a point on ellipse $C$ to the right focus is $3-2\sqrt{2}$.

$(1)$ Find the equation of ellipse $C$;  
$(2)$ Suppose a line with a slope of $-2$ intersects curve $C$ at points $E$ and $F$. Find the equation of the trajectory of the midpoint $N$ of segment $EF$;  
$(3)$ Suppose a line passing through point $F_{1}(-2\sqrt{2},0)$ intersects curve $C$ and forms a chord segment $PQ$. Find the maximum area of $\triangle PQO$ ($O$ is the origin).

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