
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 the ellipse \(\frac{x^{2}}{6}+\frac{y^{2}}{3}=1\) and a point \(P\left(1, \frac{1}{2}\right)\) inside the ellipse, draw a line through \(P\) that does not pass through the origin and intersects the ellipse at points \(A\) and \(B\). Find the maximum area of triangle \(OAB\), where \(O\) is the origin.

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