
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
 $ABCD$  is a parallelogram, and circle  $S$  (with radius  $2$ ) is inscribed insider  $ABCD$  such that  $S$  is tangent to all four line segments  $AB$ ,  $BC$ ,  $CD$ , and  $DA$ . One of the internal angles of the parallelogram is  $60^\circ$ . What is the maximum possible area of  $ABCD$ ?

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