
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
A robot vacuum cleaner called "Orthogonal" decides to clean the coordinate plane. It starts from point $O$, located at the origin of the coordinates, and travels a distance $X_{1}$ along a straight line. At the point where it arrives, the vacuum cleaner turns and describes a circle centered at the origin. After completing the circle, it continues, without changing direction, traveling a distance $X_{2}$ along a straight line. It then turns sideways towards the origin to describe another circle centered at the origin and, after completing it, travels a distance $X_{3}$. The vacuum cleaner continues this pattern. It is known that the values $X_{1}, X_{2}, X_{3}, \ldots$ follow a certain distribution with a mean $a$ and variance $d$. Find the expected value of the area of the circle described by the $n$-th circle the vacuum cleaner travels along.

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