
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 the center of the ellipse is at the origin and the right focus point F has coordinates (3, 0), the line L: x+2y-2=0 intersects the ellipse at points A and B, and the midpoint of line segment AB is $M(1, \frac{1}{2})$;

(1) Find the equation of the ellipse;
(2) If moving point N satisfies $NA \perp NB$, find the equation of the trajectory of point N.

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