
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 $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, define the "companion point" of a point $M(x_0, y_0)$ on the ellipse $C$ as $N\left(\frac{x_0}{a}, \frac{y_0}{b}\right)$.

$(1)$ Find the trajectory equation 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 $\left(\frac{1}{2}, \frac{3}{2b}\right)$, for any point $M$ on the ellipse $C$ and its "companion point" $N$, find the range of values for $\overrightarrow{OM} \cdot \overrightarrow{ON}$;

$(3)$ When $a=2$, $b= \sqrt{3}$, the line $l$ intersects the ellipse $C$ at points $A$ and $B$. If the "companion points" of points $A$ and $B$ are respectively $P$ and $Q$, 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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