
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 robotic vacuum cleaner "Orthogonal" decided to clean the coordinate plane. It starts at point $O$, located at the origin, and travels a distance $X_{1}$ along a straight line. At the point where it stops, the vacuum turns and describes a circle with its center at the origin. After completing the circle, it continues straight without changing direction for a distance $X_{2}$. Then it turns to face the origin again to describe another circle centered at the origin, and after completing this circle, it travels another straight path for a distance of $X_{3}$. This pattern continues. It is known that the distances $X_{1}, X_{2}, X_{3}, \ldots$ follow a certain distribution with an expected value $a$ and variance $d$. Find the expected value of the area of the circle bounded by the $n$-th circle described by the vacuum cleaner.

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