
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
Let \( g(x) \) be the function defined on \([-2, 2]\) by the formula 
\[
g(x) = 2 - \sqrt{4-x^2}
\]
This defines a graph of \( y = g(x) \). Consider now the graph of \( x = g(y) \). A fully enclosed region is formed by the overlay of these two graphs. Compute the area of this region.

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