
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
An $\textit{eccentric annulus}$ is a region between an outer circle and an inner circle, where the inner circle's center is shifted by a distance $d$ from the center of the outer circle. The outer circle has a radius $R$, and the inner circle has a radius $r$, with $R > r$. A line from the center of the outer circle, $O$, to a point $X$ on the outer circle serves as a radius. A tangent from $X$ to the inner circle touches it at point $Z$, and let $\overline{OY}$ be the radius of the outer circle that passes through $Z$. Define $a = XZ$, $e = XY$, and $d = O'Z$ where $O'$ is the center of the inner circle. How can we express the area of this eccentric annulus, assuming $d < R$?

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