
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 the ellipse C: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ $(a>b>0)$, the “companion point” of a point M$(x_0, y_0)$ on the ellipse C is defined as $$N\left(\frac{x_0}{a}, \frac{y_0}{b}\right)$$.  
(1) Find the equation of the trajectory of the “companion point” N of point M on the ellipse C;  
(2) If the “companion point” of the point $(1, \frac{3}{2})$ on the ellipse C is $\left(\frac{1}{2}, \frac{3}{2b}\right)$, find the range of values for $\overrightarrow{OM} \cdot \overrightarrow{ON}$ for any point M on ellipse C and its “companion point” N;  
(3) When $a=2$, $b= \sqrt{3}$, a line l intersects the ellipse C at points A and B. If the “companion points” of A and B are P and Q respectively, and the circle with diameter PQ passes through the origin O, find the area of $\triangle OAB$.

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