
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 an ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $\frac{1}{2}$, and it passes through the point $(1, \frac{3}{2})$. If point $M(x_0,y_0)$ is on the ellipse $C$, then point $N\left( \frac{x_0}{a}, \frac{y_0}{b}\right)$ is called the "ellipse point" of point $M$.  
$(1)$ Find the standard equation of ellipse $C$;  
$(2)$ If the line $l: y=kx+m$ intersects the ellipse $C$ at points $A$ and $B$, and the "ellipse points" of $A$ and $B$ are $P$ and $Q$ respectively, and the circle with $PQ$ as its diameter passes through the origin, find the area of $\triangle AOB$.

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