
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 two circles on a plane with radii of 5 and 2 and centers at points \( S_{1} \) and \( S_{2} \), respectively, touching a certain line at points \( A_{1} \) and \( A_{2} \) and lying on opposite sides of this line. The ratio of segment \( A_{1}A_{2} \) to segment \( S_{1}S_{2} \) is \( \frac{\sqrt{2}}{2} \). Find \( A_{1}A^{2} \).

In an isosceles trapezoid \( P Q R S \), the diagonals are perpendicular and divide at point \( O \) in the ratio 1 : \( \sqrt{3} \). The larger base \( P S \) of the trapezoid is equal to 1. Find the area of the common part of the circles circumscribed around triangles \( P Q O \) and \( P O S \).

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