
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
A  $ 1$  by  $ 1$  square  $ABCD$  is inscribed in the circle  $m$ . Circle  $n$  has radius  $1$  and is centered around  $A$ . Let  $S$  be the set of points inside of  $m$  but outside of  $n$ . What is the area of  $S$ ?

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