
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 $xOy$, point $F$ is a focus of the ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, and point $B_1(0, -\sqrt{3})$ is a vertex of $C$, $\angle OFB_1 = \frac{\pi}{3}$.

$(1)$ Find the standard equation of $C$;

$(2)$ If point $M(x_0, y_0)$ is on $C$, then point $N(\frac{x_0}{a}, \frac{y_0}{b})$ is called an "ellipse point" of point $M$. The line $l$: $y = kx + m$ intersects $C$ at points $A$ and $B$, and the "ellipse points" of $A$ and $B$ are $P$ and $Q$ respectively. If the circle with diameter $PQ$ passes through point $O$, find the area of $\triangle AOB$.

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