
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
From each vertex of a square with side length $2a$, quarter circles with radius $a$ are drawn within the square. A new square is inscribed inside the resulting curved quadrilateral, with sides tangent to these quarter circles. The area of this inscribed square depends on the orientation of its sides. Investigate and graph the variation of this square's area based on this orientation.

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