
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
Four circles with a radius of 1 and centers at \( A, B, C, D \) are placed on a plane so that each circle touches exactly two others. A fifth circle touches two of these circles externally and passes through the centers of the other two. Find all possible values of the area of quadrilateral \( ABCD \).

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