
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 $E$: $\frac{x^2}{4}+\frac{y^2}{3}=1$ with two foci $F_{1}$ and $F_{2}$, a line $l: y=kx+m$ where $k, m \in \mathbb{R}$ intersects the ellipse at points $A$ and $B$.<br/>$(1)$ If the line $l$ passes through the point $C(0,3)$ and $|OA|=|AC|$, find the coordinates of point $A$;<br/>$(2)$ If the line $l$ passes through the point $C(0,3)$ and $S_{\triangle AOC}=S_{\triangle AOB}$, find the equation of the line $l$;<br/>$(3)$ If $k_{OA} \cdot k_{OB}=-\frac{3}{4}$, is the area of $\triangle AOB$ a constant value? If yes, find the constant value; if not, explain the reason.

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