
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 a rectangular coordinate system, \( O \) is the origin, and points \( A \) and \( B \) are in the first quadrant. \( A \) lies on the line \( y = \tan \theta x \), where \( \theta \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right) \) and \( |OA| = \frac{1}{\sqrt{2} - \cos \theta} \). Point \( B \) is on the hyperbola \( x^2 - y^2 = 1 \) such that the area of triangle \( \triangle OAB \) is minimized. Determine the value of \( \theta \) within \( \left( \frac{\pi}{4}, \frac{\pi}{2} \right) \) that maximizes the area of \( \triangle OAB \), and find the maximum area.

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