
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 centered at the origin \( O \) with foci on the \( x \)-axis, and an eccentricity of \( \frac{\sqrt{3}}{2} \), which also passes through the point \( \left(\sqrt{2}, \frac{\sqrt{2}}{2}\right) \). Assume a line \( l \) not passing through the origin \( O \) intersects the ellipse at points \( P \) and \( Q \), and the slopes of the lines \( OP \), \( PQ \), and \( OQ \) form a geometric sequence. Determine the range of the area of the triangle \( \triangle OPQ \).

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