
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 points $M$ and $N$ are respectively the left and right vertices of the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, and $F$ is its right focus. The geometric mean of $|MF|$ and $|FN|$ is $\sqrt{3}$, and the eccentricity of the ellipse is $\frac{1}{2}$.
$(1)$ Find the equation of ellipse $C$;
$(2)$ Suppose a line $l$ that does not pass through the origin $O$ intersects the trajectory at points $A$ and $B$. If the slopes of lines $OA$, $AB$, and $OB$ form a geometric sequence in order, find the range of the area of $\triangle OAB$.

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