
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
Let \( AB \) be a line segment with length 2, and \( S \) be the set of points \( P \) on the plane such that there exists a point \( X \) on segment \( AB \) with \( AX = 2 \cdot PX \). Find the area of \( S \).

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