
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
In the given diagram, there is a blueprint of a square window. Four congruent circles each pass through one vertex of a small square, potentially touching adjacent circles. The four circular regions and the small square region are shaded, while the rest of the large square is transparent. What should be the side length of the small square so that the total area of the shaded regions is a) minimal b) maximal?

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