
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

The robot vacuum "Orthogonal" decided to clean the coordinate plane. It starts from point \( O \), located at the origin, and moves straight for a distance \( X_1 \). At the point where it arrives, the vacuum turns and traces a circle centered at the origin. After completing the circle, it, without changing direction, moves straight for a distance \( X_2 \). Then it again turns sideways towards the origin to trace another circle centered at the origin, and after completing it, moves straight for a distance \( X_3 \). This process continues similarly. It is known that the distances \( X_1, X_2, X_3, \ldots \) follow some distribution with an expectation \( a \) and variance \( d \). Find the expected area of the circle circumscribed by the \( n \)-th circle traced by the vacuum.

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