
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 "Orthogonal" wants 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 it reaches, the vacuum turns and describes a circle centered at the origin. After completing the circle, it travels a distance $X_{2}$ in a straight line without changing direction. Then it turns to face sideways to the origin to describe another circle centered at the origin and proceeds similarly with distance $X_{3}$. This process continues further. The distances $X_{1}, X_{2}, X_{3}, \ldots$ follow some 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 that the vacuum cleaner follows.

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