
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 diagram shows a circle with radius 2 and a square with sides of length 2. The center of the circle lies on the perpendicular bisector of a side of the square, at a distance \( x \) from the side, as shown. The shaded region - inside the square but outside the circle - has an area of 2. What is the value of \( x \)?
$$
\begin{array}{l}
\text { A } \frac{\pi}{3}+\frac{\sqrt{3}}{2}-1 \\
\text { B } \frac{\pi}{3}+\frac{\sqrt{3}}{4}-1 \\
\text { C } \frac{\pi}{3}+\frac{1}{2} \\
\text { D } \frac{\pi}{3}+1 \\
\text { E } \frac{\pi}{3} \\
\end{array}
$$

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