
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
An isosceles triangle PQR is inscribed in the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ such that vertex Q is at (0, b), and base PR is parallel to the x-axis. Additionally, vertex P lies on the ellipse at a horizontal distance 2c from the y-axis (i.e., P is at (c, y) for some y). Base angle $\theta$ at P and R with respect to the vertical line through Q is known. Find the area of the segment of the ellipse that lies above triangle PQR, assuming the ellipse's semi-major axis a is 3, the semi-minor axis b is 2, and c = 1.

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