
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 cube \( ABCD-A_{1}B_{1}C_{1}D_{1} \) with \( AA_{1} = 1 \), \( E \) and \( F \) being the midpoints of edges \( CC_{1} \) and \( DD_{1} \) respectively, what is the area of the cross-section of the circumscribed sphere of the cube cut by plane \( AEF \)?

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